As in the example of $C_2(R) = C(R, \emptyset; \Theta)$, for a trivalent Jacobi diagram $\Gamma$, our general compactification $C(R, \emptyset; \Gamma)$ of the space $\tilde{C}(R, \emptyset; \Gamma) = \tilde{C}_{V(\Gamma)}(R)$ of injective maps from $V(\Gamma)$ to $\tilde{R}$ depends only on the finite set $V = V(\Gamma)$ of vertices of $\Gamma$. It is denoted by $C_V(R)$. As in the example of $C(S^3, K; \text{↔})$, for a Jacobi diagram $\Gamma$ on the domain of a link $L$ of $\tilde{R}$, we define the compactification $C(R, L; \Gamma)$ of $\tilde{C}(R, L; \Gamma)$ to be the closure of $\tilde{C}(R, L; \Gamma)$ in $C_{V(\Gamma)}(R)$ in Proposition 8.6. This is why we first study $C_V(R)$ by generalizing the construction of $C_2(R)$ performed in Section 3.2. In this general case, we start with $R^V$ and blow up all the diagonals and all the loci that involve $\infty$, in the sense of Section 3.1, following a process precisely described in Theorem 8.4.

In this informal introduction, we forget about $\infty$ and first discuss how we successively blow up the diagonals in the manifold $(\mathbb{R}^3)^V$ of maps from $V$ to $\mathbb{R}^3$. Thus, we get the preimage $C_V[\mathbb{R}^3]$ of $(\mathbb{R}^3)^V$ in $C_{V(\Gamma)}(S^3)$ under the composition valued in $(S^3)^V$ of the blowdown maps.

### 8.1.1 On the configuration space $C_V[\mathbb{R}^3]$ of four points in $\mathbb{R}^3$

The diagonal $\Delta_V((\mathbb{R}^3)^V)$ is the set of constant maps from $V$ to $\mathbb{R}^3$. The fiber of the normal bundle to the vector space $\Delta_V((\mathbb{R}^3)^V)$ in $(\mathbb{R}^3)^V$ is the vector space $(\mathbb{R}^3)^V/\Delta_V((\mathbb{R}^3)^V)$. It consists of maps $c$ from $V$ to $\mathbb{R}^3$ up to global translation.<fnref n="1" /> The fiber of the unit normal bundle to the diagonal $\Delta_V((\mathbb{R}^3)^V)$ is the space $\bar{S}_V(\mathbb{R}^3)$ of nonconstant maps from $V = \{v_1, v_2, \dots, v_{|V|}\}$ to $\mathbb{R}^3$ up to (global) translation and up to dilation.<fnref n="2" /> The space $\bar{S}_V(\mathbb{R}^3)$, studied in Section 8.3, can be identified with the space of maps $w: V \to \mathbb{R}^3$ that map $v_1$ to 0 and such that $\sum_{i=2}^{|V|} \|w(v_i)\|^2 = 1$. It is diffeomorphic to a sphere of dimension $(3|V| - 4)$. Let $B\ell_1 = B\ell((\mathbb{R}^3)^V, \Delta_V((\mathbb{R}^3)^V))$ be obtained from $(\mathbb{R}^3)^V$ by blowing up $\Delta_V((\mathbb{R}^3)^V)$. We have a diffeomorphism

$$ \psi: \mathbb{R}^3 \times [0, \infty[ \times \bar{S}_V(\mathbb{R}^3) \to B\ell_1, $$

which maps $(u, \mu, w)$ to the map $c: V \to \mathbb{R}^3$ such that $c(v_i) = u + \mu w(v_i)$ for any $i \in |V|$, with respect to the identification above. This map $c$ is furthermore equipped with the data of the map $w$ when $c$ is constant, or equivalently when $\mu = 0$. In particular, this first blow-up equips each constant map $c_0$ in the manifold $B\ell_1$ with the additional data of a nonconstant map $w: V \to \mathbb{R}^3$ up to translation and dilation. Let $c_0$ be the constant map which maps $V$ to $u$, and let $(c_0, w)$ denote $\psi(u, 0, w)$, then $(c_0, w)$ is the limit in $B\ell_1$ of

<footnote n="1">¹A map $c$ is identified with the map $(v \mapsto c(v) + W)$ for any $W \in \mathbb{R}^3$.</footnote>

<footnote n="2">²In $\bar{S}_V(\mathbb{R}^3)$, a map $c$ is furthermore identified with the map $(v \mapsto \lambda c(v))$ for any $\lambda \in ]0, +\infty[$.</footnote>
$\psi(u, t, w)$, when $t > 0$ tends to 0. Therefore, we can think of the map $w$ as an infinitesimal configuration. The first blow-up provides a magnifying glass, allowing us to see this infinitesimal configuration $w$ of the vertices mapped to the same point $u$ in $\mathbb{R}^3$.

When $|V| = 2$, we are done and $B\ell_1$ is the preimage $C_V[\mathbb{R}^3]$ of $(\mathbb{R}^3)^2$ in $C_2(S^3)$. In general, we blow up the other diagonals $\Delta_A((\mathbb{R}^3)^V)$ for all subsets $A$ of $V$, where $\Delta_A((\mathbb{R}^3)^V)$ is the subspace of $(\mathbb{R}^3)^V$ consisting of the maps $c$ that map $A$ to a single element and such that $c(V \setminus A) \subset \mathbb{R}^3 \setminus c(A)$, as in Section 8.2.

Let us describe the process when $V = V(\Gamma) = \{v_1, v_2, v_3, v_4\}$. From now on, we restrict to this case in this subsection. The closure of the diagonal $\Delta_A((\mathbb{R}^3)^V)$ in $B\ell_1$ is $\psi(\mathbb{R}^3 \times [0, \infty[ \times \Delta_A(\overline{\mathcal{S}}_V(\mathbb{R}^3)))$, where $\Delta_A(\overline{\mathcal{S}}_V(\mathbb{R}^3))$ is the subspace of $\overline{\mathcal{S}}_V(\mathbb{R}^3)$ made of the classes of its maps $c$ constant on $A$ such that $c(V \setminus A) \subset \mathbb{R}^3 \setminus c(A)$. In particular, the closures of the diagonals $\Delta_A((\mathbb{R}^3)^V)$ in $B\ell_1$ for the subsets $A$ of $V$ of cardinality 3 are disjoint in $B\ell_1$. We blow up these closures in an arbitrary order. Since the closures are disjoint, the order of the blow-ups does not affect the result.

The fiber of the unit normal bundle of $\Delta_A((\mathbb{R}^3)^V)$ in $B\ell_1$ is the space $\overline{\mathcal{S}}_A(\mathbb{R}^3)$ of nonconstant maps from $A$ to $\mathbb{R}^3$ up to (global) translation and dilation. Let $B = A_{123} = \{v_1, v_2, v_3\}$. View $\overline{\mathcal{S}}_{123}(\mathbb{R}^3) = \overline{\mathcal{S}}_B(\mathbb{R}^3)$ as the space of maps $w_{123}: V \to \mathbb{R}^3$ that map $v_1$ and $v_4$ to 0 and such that $\|w_{123}(v_2)\|^2 + \|w_{123}(v_3)\|^2 = 1$. We have a smooth embedding

$$ \psi_2: \mathbb{R}^3 \times [0, \infty]^2 \times S^2 \times \overline{\mathcal{S}}_B(\mathbb{R}^3) \to B\ell(B\ell_1, \overline{\Delta_B((\mathbb{R}^3)^V)}) $$

that maps $(u, \mu, \mu_{123}, W_4, w_{123})$ to the map $c: V \to \mathbb{R}^3$ such that $c(v_4) = u + \frac{\mu}{\sqrt{1+\mu_{123}^2}}W_4$ and $c(v_i) = u + \frac{\mu}{\sqrt{1+\mu_{123}^2}}\mu_{123}w_{123}(v_i)$ for $i \in \underline{3}$. The preim- age of $\overline{\Delta_B((\mathbb{R}^3)^V)}$ under the blowdown map $B\ell(B\ell_1, \overline{\Delta_B((\mathbb{R}^3)^V)}) \to B\ell_1$ is $\psi_2(\mathbb{R}^3 \times [0, \infty[ \times \{0\} \times S^2 \times \overline{\mathcal{S}}_B(\mathbb{R}^3))$, where $\mu_{123} = 0$. The image of $\psi_2$ contains a neighborhood of this preimage in $B\ell(B\ell_1, \overline{\Delta_B((\mathbb{R}^3)^V)})$.

Here, this blow-up equips a map $c$ of $\Delta_B((\mathbb{R}^3)^V)$ with the additional data of the (infinitesimal, nonconstant) configuration $w_{123}|_B: B \to \mathbb{R}^3$ up to translation and dilation. It equips a constant map $c_0$ with value $u$ in the closure of $\Delta_B((\mathbb{R}^3)^V)$ in $B\ell_1$ with such a configuration $w_{123}|_B$ in addition to the former $w$, which maps $B$ to 0 and $v_4$ to some $W_4 \in S^2$. In this case, we denote the obtained configuration $\psi_2(u, 0, 0, W_4, w_{123})$ by $(c, w, w_{123}|_B)$, and we have three observation scales. All the $c(v_i)$ coincide at the first initial scale. At the second (infinitely smaller) scale $w$ in $B\ell_1$, the $w(v_i)$ coincide for $i \in \underline{3}$, but $(w(v_4) - w(v_1))$ is not zero, and its direction is a vector $W_4 \in S^2$. At the third scale (infinitely smaller than the second one) in $B\ell(B\ell_1, \overline{\Delta_B((\mathbb{R}^3)^V)})$, the configuration $w_{123}|_B$ of $\{v_1, v_2, v_3\}$ is visible up to
global translation and dilation. The first three pictures of Figure 8.1 show these three observation scales.

<figure><img src="image_72.png" /><figcaption>Figure 8.1: The magnifying glasses provided by the successive blow-ups from $(\mathbb{R}^3)^V$ to $C_V[\mathbb{R}^3]$, for a configuration $(c, w, w_{123}, w_{23})$</figcaption></figure>

Let $B\ell_2$ be the manifold obtained by blowing up the four closures of the diagonals $\Delta_A((\mathbb{R}^3)^V)$ in $B\ell_1$ for the subsets $A$ of $V$ of cardinality 3. We have local charts similar to $\psi_2$ for $B\ell_2$ in the neighborhoods of the disjoint blown-up loci.

Finally, we blow up (the preimages under the composition of the previous blowdown maps of) the closures of the diagonals $\Delta_A((\mathbb{R}^3)^V)$ in $B\ell_2$ for the subsets $A$ of $V$ of cardinality 2, in an arbitrary order, to get the manifold $C_V[\mathbb{R}^3]$ of Section 8.6. Here the diagonals $\Delta_{\{v_1,v_2\}}((\mathbb{R}^3)^V)$ and $\Delta_{\{v_3,v_4\}}((\mathbb{R}^3)^V)$ are no longer disjoint. Nevertheless, the blow-up operations associated to $A_{12} = \{v_1, v_2\}$ and $A_{34} = \{v_3, v_4\}$, which act on different coordinates, commute. In the neighborhood of the intersection of the corresponding blown-up loci in $C_V[\mathbb{R}^3]$, we have an embedding

$$ \psi_3: \mathbb{R}^3 \times [0, \infty[ \times [0, 1/3]^2 \times (S^2)^3 \to C_V[\mathbb{R}^3], $$

which maps

$$ (u, \mu', \mu_{12}, \mu_{34}, W_3, W_{12}, W_{34}) $$

to the map $c: V \to \mathbb{R}^3$ such that $c(v_1) = u$, $c(v_2) = u + \mu'\mu_{12}W_{12}$, $c(v_3) = u + \mu'W_3$, $c(v_4) = u + \mu'(W_3 + \mu_{34}W_{34})$ where $W_3 \in S^2 = \bar{S}_{\{A_{12},A_{34}\}}(\mathbb{R}^3)$,
$W_{12} \in S^2 = \bar{S}_{A_{12}}(\mathbb{R}^3)$, and $W_{34} \in S^2 = \bar{S}_{A_{34}}(\mathbb{R}^3)$. The configuration $c$ is equipped with the map $w$ up to translation and dilation when $c$ is constant. (A representative $w$ of this map sends the elements of $V$ to $w(v_1) = 0$, $w(v_2) = \mu_{12}W_{12}$, $w(v_3) = W_3$, and $w(v_4) = W_3 + \mu_{34}W_{34}$.) The configuration $c$ is equipped with $W_{12}$ when the restriction of $w$ (or $c$) to $A_{12}$ is constant. It is equipped with $W_{34}$, when the restriction of $w$ (or $c$) to $A_{34}$ is constant. Figure 8.2 shows the three magnifying glasses that have popped up for a configuration $(c, w, W_{12}, W_{34}) = \psi_3(u, 0, 0, 0, W_3, W_{12}, W_{34})$.

<figure><img src="image_73.png" /><figcaption>Figure 8.2: The magnifying glasses provided by the successive blow-ups from $(\mathbb{R}^3)^V$ to $C_V[\mathbb{R}^3]$, for a configuration $(c, w, W_{12}, W_{34})$</figcaption></figure>

For a constant map $c$, whose associated infinitesimal $w$ is constant on $A_{123}$, and whose next associated $w_{123}$ is constant on $A_{23} = \{v_2, v_3\}$, the third blow-up family provides a fourth smaller observation scale. With this additional scale, we see a nonconstant map $w_{23}: A_{23} \to \mathbb{R}^3$ up to global translation and dilation, as in Figure 8.1. The map $w_{23}$ gives the direction from $w_{23}(v_2)$ to $w_{23}(v_3)$.

### 8.1.2 More configuration spaces and their stratifications

In general, for a finite set $V$ and for an integer $d \in \mathbb{N} \setminus \{0\}$, we transform $(\mathbb{R}^d)^V$ to a manifold $C_V[\mathbb{R}^d]$ by successively blowing up the closures of (the preimages under the composition of the previous blowdown maps of) the diagonals $\Delta_A((\mathbb{R}^d)^V)$ associated to the subsets $A$ of $V$ of cardinality $k$, for $k = |V|, |V|-1, \dots, 2$ in this decreasing order, in Section 8.6. It will follow from Theorem 8.32 and Proposition 8.33 that there are natural smooth (quotients of) restriction maps from $C_V[\mathbb{R}^d]$ to the space $\bar{S}_A(\mathbb{R}^d)$ of nonconstant maps from $A$ to $\mathbb{R}^d$ up to translation and dilation, for every subset $A$ of $V$ of cardinality at least 2. These restriction maps are smooth extensions of the natural (quotients of) restriction maps from the space $\tilde{C}_V[\mathbb{R}^d]$ of injective maps from $V$ to $\mathbb{R}^d$ to the compact space $\bar{S}_A(\mathbb{R}^d)$. We could define the
configuration space $C_V[\mathbb{R}^d]$ to be the closure of the image of $\tilde{C}_V[\mathbb{R}^d]$ in the product $(\mathbb{R}^d)^V \times \prod_{A \subseteq V; |A| \ge 2} \bar{S}_A(\mathbb{R}^d)$, or in smaller spaces, as Dev Sinha did in [Sin04]. This would define its topology. However, the differential structures of our configuration spaces are essential for our purposes. This is why we study them in detail in this chapter.

**Definition 8.1.** The partition associated to a map $f$ from a finite set $V$ to some set $X$ is the following set $K(V; f)$ of subsets of $V$.

$$K(V; f) = \{f^{-1}(x); x \in f(V)\}.$$

In this book, a partition of a finite set $V$ is a set of disjoint nonempty subsets of $V$ whose union is $V$. The elements of a partition $K(V)$ are called the kids of $V$ (with respect to the partition). (We do not call them children because we use the initial of children in the notation of configuration spaces.) The daughters of $V$ with respect to such a partition are its kids with cardinality at least 2, and its sons are the singletons of $K(V)$. (The daughters might bear smaller kids in other partitions.) We denote the set of daughters of $V$ by $D(V, K(V))$, or by $D(V)$ when $K(V)$ is understood. We simply denote $D(V, K(V; f))$ by $D(V; f)$.

**Definition 8.2.** A parenthesization $\mathcal{P}$ of a finite set $V$ is a set $\mathcal{P} = \{A_i; i \in I\}$ of subsets of $V$, each of cardinality greater than one, such that, for any two distinct elements $i, j$ of $I$, one of the following holds $A_i \subset A_j$, $A_j \subset A_i$ or $A_i \cap A_j = \emptyset$.

Every element $x$ of the space $C_V[\mathbb{R}^d]$ defines the following parenthesization $\mathcal{P}(x)$ of $V$. The maximal elements (with respect to the inclusion) of $\mathcal{P}(x)$ are the daughters of $V$ with respect to $(p_b(x) \in (\mathbb{R}^d)^V)$. For any element $A$ of $\mathcal{P}(x)$, the maximal strict subsets of $A$ in $\mathcal{P}(x)$ are the daughters of $A$ with respect to the restriction of $x$ to $A$. In our examples of Figures 8.1 and 8.2, the parenthesizations are $\{V, A_{123}, A_{23}\}$ and $\{V, A_{12}, A_{34}\}$, respectively. They are in one-to-one correspondences with the magnifying glasses provided by the iterated blow-ups, or, equivalently, with the blow-ups that affected $x$. For a parenthesization $\mathcal{P}$ of $V$, define the *stratum* $C_{V,\mathcal{P}}[\mathbb{R}^d] = \{x \in C_V[\mathbb{R}^d] : \mathcal{P}(x) = \mathcal{P}\}$. As we can see in the above examples, and as we will state in a larger generality in Proposition 8.34, such a stratum is a smooth manifold of codimension $|\mathcal{P}|$ in $C_V[\mathbb{R}^d]$. As a set, the configuration space $C_V[\mathbb{R}^d]$ is the disjoint union of the strata $C_{V,\mathcal{P}}[\mathbb{R}^d]$. So these strata define a *stratification* of $C_V[\mathbb{R}^d]$. The open codimension-one faces of $C_V[\mathbb{R}^d]$ are the strata of codimension one. They are in one-to-one
correspondence with the subsets $A$ of $V$ of cardinality greater than 1. The open face associated to $A$ consists of the pairs $(c \in \Delta_A((\mathbb{R}^d)^V), w_A \in \bar{S}_A(\mathbb{R}^d))$ such that the restriction of $c$ to $(V \setminus A) \cup \{a \in A\}$ and the map $w_A$ are injective.

The space $\bar{S}_V(\mathbb{R}^d)$ of nonconstant maps from $V$ to $\mathbb{R}^d$ up to translation and dilation is the preimage of a constant map under the blowdown map from $\text{Bl}((\mathbb{R}^d)^V, \Delta_V((\mathbb{R}^d)^V))$ to $(\mathbb{R}^d)^V$. In Theorem 8.11, we successively blow up the diagonals $\Delta_A(\bar{S}_V(\mathbb{R}^d))$ of this space, for strict subsets $A$ of $V$ of cardinality greater than 1, as above, to transform $\bar{S}_V(\mathbb{R}^d)$ to the preimage $S_V(\mathbb{R}^d)$ of a constant map under the composed blowdown map from $C_V[\mathbb{R}^d]$ to $(\mathbb{R}^d)^V$. The space $S_V(\mathbb{R}^d)$ is presented in detail in Section 8.3. It is a compactification of the space $\dot{S}_V(\mathbb{R}^d)$ of injective maps from $V$ to $\mathbb{R}^d$ up to translation and dilation.

The spaces $\tilde{S}_2(\mathbb{R}^d)$, $\bar{S}_2(\mathbb{R}^d)$, $S_2(\mathbb{R}^d)$ are identical, they consist of the classes of the maps $w_X: \underline{2} \to \mathbb{R}^d$ such that $w_X(1) = 0$ and $w_X(2) = X$ for $X \in S^{d-1}$. So they are diffeomorphic to the unit sphere $S^{d-1}$ of $\mathbb{R}^d$. In particular, the space $\bar{S}_2(\mathbb{R})$ has two elements, which are the classes of $w^+$ and $w^-$, where $w^+$ and $w^-$ map 1 to 0, $w^+(2) = 1$, and $w^-(2) = -1$.

Let us discuss the spaces $\tilde{S}_V(\mathbb{R})$, $\bar{S}_V(\mathbb{R})$, and $S_V(\mathbb{R})$ in more detail.

**Example 8.3.** In general, for an integer $k \ge 2$, $\tilde{S}_k(\mathbb{R})$ and its compactification $S_k(\mathbb{R})$ have $k!$ connected components, which correspond to the orders of the $c(i)$ in $\mathbb{R}$, for configurations $c$ in $\tilde{S}_k(\mathbb{R})$. Denote the connected component of $\tilde{S}_k(\mathbb{R})$ in which the configurations $c$ satisfy $c(1) < c(2) < \cdots < c(k)$, by $\tilde{S}_{<,k}(\mathbb{R})$. Its respective closures in $S_k(\mathbb{R})$ and in $\bar{S}_k(\mathbb{R})$ are denoted by $S_{<,k}(\mathbb{R})$ and $\bar{S}_{<,k}(\mathbb{R})$. Then we have $\tilde{S}_{<,\underline{3}}(\mathbb{R}) = \{(0, t, 1) : t \in [0, 1]\}$. The spaces $S_{<,\underline{3}}(\mathbb{R})$ and $\bar{S}_{<,\underline{3}}(\mathbb{R})$ coincide. They are the natural compactification $[0, 1]$ of $\tilde{S}_{<,\underline{3}}(\mathbb{R})$. In this space, an element $t$ of $]0, 1[$ represents the injective configuration $(0, t, 1)$. In $[0, 1]$, the extremity 0 represents the limit configuration $((..)) = \lim_{t \to 0}(0, t, 1)$ and 1 represents the limit configuration $(..)) = \lim_{t \to 0}(0, 1-t, 1)$. The configuration space $S_{<,\underline{4}}(\mathbb{R})$ is diffeomorphic to the following well-known pentagon:

<figure><img src="image_74.png" /></figure>

The edge from $((..))$ to $((..))$ is the preimage of the diagonal $\Delta_{\underline{2}}(\bar{S}_{<,V}(\mathbb{R}))$ under the composed blowdown map from $S_{<,\underline{4}}(\mathbb{R})$ to $\bar{S}_{<,\underline{4}}(\mathbb{R})$. It is naturally diffeomorphic to $S_{<,\underline{3}}(\mathbb{R})$. The edge from $((..))$ to $((..))$ is the closure of the preimage of the diagonal $\Delta_{\underline{2}}(\bar{S}_{<,V}(\mathbb{R}))$. Its interior is naturally diffeomorphic to $\tilde{S}_{<,\{2,3,4\}}(\mathbb{R})$.
In general, for $k \ge 3$, the configuration space $S_{<,k}(\mathbb{R})$ is a Stasheff polyhedron [Sta63] of dimension $(k-2)$ whose corners (i.e., ridges of dimension 0) are labeled by nonassociative words in the letter $\bullet$, as in the above example. For any integer $k \ge 2$, a nonassociative word $w$ with $k$ letters represents a limit configuration $w = \lim_{t \to 0} w(t)$, where $w(t) = (w_1(t) = 0, w_2(t), \dots, w_{k-1}(t), w_k(t) = 1)$ is an injective configuration inductively defined as follows for $t \in [0, \frac{1}{2}]$. If $w$ is the product $uv$ of a nonassociative word $u$ of length $j \ge 1$ and a nonassociative word $v$ of length $(k-j) \ge 1$, then $w_i(t) = tu_i(t)$ when $1 < i \le j$, and $w_i(t) = 1-t+tv_{i-j}(t)$ when $j < i < k$. For example, we have (((..).).)(t) = (0, t², t, 1). In a limit configuration associated to such a nonassociative word, points inside matching parentheses are thought of as infinitely closer to each other than they are to points outside these matching parentheses. The parenthesization associated as above to a nonassociative word is the set of strict subsets inside matching parentheses.<fnref n="3" />

## 8.2 General presentation of $C_V(R)$

Let $V$ denote a finite set. We use this notation for a generic finite set since our sets will end up being sets of vertices of Jacobi diagrams. The space of maps from $V$ to $X$ is denoted by $X^V$ as usual. For a subset $A$ of $V$ of cardinality at least 2, recall that the subspace of $X^V$ consisting of maps $c$ that map $A$ to a single element and such that $c(V \setminus A) \subset X \setminus c(A)$ is a diagonal denoted by $\Delta_A(X^V)$. In particular, if $|V| \ge 2$, the small diagonal consisting of constant maps is denoted by $\Delta_V(X^V)$.

As often in this book, we fix a rational homology sphere $R$ and a point $\infty$ of $R$. Recall that $\tilde{C}_V(R)$ is the space of injective maps from $V$ to $(\tilde{R} = R \setminus \{\infty\})$.

**Theorem 8.4.** Define a compactification $C_V(R)$ of $\tilde{C}_V(R)$ as follows. For a nonempty $A \subseteq V$, let $\Xi_A$ be the set of maps from $V$ to $R$ that map $A$ to $\infty$ and $V \setminus A$ to $R \setminus \{\infty\}$. Start with $R^V$. Blow up $\Xi_V$ (which is reduced to the point $m = \infty^V$ such that $m^{-1}(\infty) = V$). Set

$$C_{V,|V|+1}(R) = \mathcal{B}(R^V, \Xi_V).$$

For $k = |V|, |V| - 1, \dots, 3, 2$, let $C_{V,k}(R)$ be obtained from $C_{V,k+1}(R)$ by blowing up the closures of (the preimages under the composition of the previous blowdown maps of) the $\Delta_A(\tilde{R}^V)$ such that $|A| = k$ and the closures of (the preimages under the composition of the previous blowdown maps of)

<footnote n="3">³In Theorem 8.28, we will rather associate the set of all subsets inside matching parentheses.</footnote>
the $\Xi_J$ such that $|J| = k - 1$. At each step, the blown-up manifolds are smooth and transverse to the ridges, and $C_{V,k}(R)$ is independent of the order of the blow-ups. The obtained manifold $C_V(R) = C_{V,2}(R)$ is a smooth compact (3$|V|$)-manifold with ridges. The interior of $C_V(R)$ is $\tilde{C}_V(R)$, and the composition of the blowdown maps gives rise to a canonical smooth blowdown projection $p_b: C_V(R) \to R^V$.

We prove the generalization Theorem 8.35 of Theorem 8.4 in Section 8.7 using the results of Sections 8.5 and 8.6. This generalization includes an alternative definition of $C_V(R)$.

Set $C_n(R) = C_n(\tilde{R})$.

As already announced, with the above definition, we have $C_1(R) = \mathcal{B}(R, \infty)$, and the configuration space $C_2(R)$ is the compactification studied in Section 3.2. In particular, Theorem 8.4 is true when $|V| \le 2$.

**Theorem 8.5.** The configuration spaces $\tilde{C}_V(R)$ have the following properties.

1. Under the assumptions of Theorem 8.4, for $A \subset V$, the restriction map

$$p_A: \tilde{C}_V(R) \to \tilde{C}_A(R)$$

extends to a smooth restriction map still denoted by $p_A$ from $C_V(R)$ to $C_A(R)$ such that the following square commutes:

$$\begin{tikzcd}
C_V(R) \arrow[r, "p_A"] \arrow[d, "p_b"] & C_A(R) \arrow[d, "p_b"] \\
R^V \arrow[r, "p_A"] & R^A
\end{tikzcd}$$

2. For an open subset $U$ of $R$, let $C_V(U)$ denote $p_b^{-1}(U^V)$. If $V = \cup_{i \in I} A_i$ and if $(U_i)_{i \in I}$ is a family of disjoint open sets of $R$, then the product

$$p_b^{-1}\left(\prod_{i \in I} U_i^{A_i}\right) \xrightarrow{\prod_{i \in I} p_{A_i}} \prod_{i \in I} C_{A_i}(U_i)$$

of the above restriction maps is a diffeomorphism.

The first part of this theorem will be a direct consequence of Proposition 8.33 and Theorem 8.35. Its second part comes from the locality of the blow-up operations. The spaces $C_V(\tilde{R})$, which involve only blow-ups along the diagonals, have been studied by Scott Axelrod and Isadore Singer [AS94, Section 5], and with more details by Dev Sinha [Sin04]. Their properties that are useful in this book are proved in Sections 8.5 and 8.6. William Fulton
and Robert MacPherson studied similar compactifications of configuration spaces in an algebraic geometry setting in [FM94].

Recall the configuration space

$$ \check{C}(R, L; \Gamma) = \{c: U \cup T \hookrightarrow \check{R} : (\exists j \in [i_{\Gamma}] : c|_{U} = L \circ j)\} $$

of Section 7.1.

**Proposition 8.6.** The closure of $\check{C}(R, L; \Gamma)$ in $C_{V(\Gamma)}(R)$ is a smooth compact submanifold of $C_{V(\Gamma)}(R)$ transverse to the ridges. It is denoted by $C(R, L; \Gamma)$.

We will prove Proposition 8.6 in Section 8.8. Theorems 8.4 and 8.5, and Proposition 8.6 imply Proposition 7.8.

## 8.3 Configuration spaces associated to unit normal bundles to diagonals

For a vector space $T$, recall from Section 3.1 that $S(T)$ denotes the quotient $S(T) = (T \setminus \{0\})/\mathbb{R}^{+*}$ of $T \setminus \{0\}$ by the dilations. If $T$ is equipped with a Euclidean norm, then $\mathbb{S}(T)$ denotes the unit sphere of $T$. In this case, $S(T)$ and $\mathbb{S}(T)$ are diffeomorphic.

**Definition 8.7.** Let $V$ denote a finite set of cardinality at least 2. We use the notation $T$ for a generic vector space since $T$ will end up being a tangent space. Let

$$ \bar{S}_V(T) = S(T^V / \Delta_V(T^V)) = \frac{(T^V / \Delta_V(T^V)) \setminus \{0\}}{\mathbb{R}^{+*}} $$

be the space of nonconstant maps from $V$ to $T$ up to translation and dilation.

**Lemma 8.8.** Let $A$ be a subset of $V$. The fiber of the unit normal bundle to $\Delta_A(R^V)$ in $R^V$ over a configuration $m$ is $\bar{S}_A(T_{m(A)}R)$.

**PROOF:** Exercise. $\square$

<figure><img src="image_75.png" /></figure>

As expected for the fiber of a unit normal bundle, we have the following lemma.

**Lemma 8.9.** The topological space $\bar{S}_V(T)$ has a canonical smooth structure. When $T$ is a vector space of dimension $\delta$, the space $\bar{S}_V(T)$ is diffeomorphic to a sphere of dimension $((|V| - 1)\delta - 1)$.
PROOF: Choosing a basepoint $b(V)$ for $V$ and a basis for $T$ identifies $\bar{S}_V(T)$ with the set $\mathbb{S}(T^{V\setminus\{b(V)\}})$ of maps $c: V \to T$ such that

* $c(b(V)) = 0$ and

* $\sum_{v \in V} \|c(v)\|^2 = 1$,

with respect to the Euclidean norm for which our basis is orthonormal. It is easy to see that changes of basepoints, or bases of $T$ give rise to diffeomorphisms of spheres. $\square$

**Convention 8.10.** In this chapter, we do not take care of orientations. Later, we will associate the following orientation of $\bar{S}_V(T)$ to an order of $V$ and an orientation of $T$. Assume that $V = \{v_1, \dots, v_n\}$ and that $T$ is oriented. The order on $V$ orients $T^V$ and $T^{V\setminus\{v_1\}}$. Then the map from the boundary of the unit ball of $T^{V\setminus\{v_1\}}$ to $\bar{S}_V(T)$ that maps an element $(x_2, \dots, x_n)$ of $\mathbb{S}(T^{V\setminus\{v_1\}})$ to the class of $(0, x_2, \dots, x_n)$ is an orientation-preserving diffeomorphism.

When $V = \underline{2}$, fixing the basepoint $b(\underline{2})$ of $\underline{2}$ to be 1 identifies $\bar{S}_V(T)$ with the unit sphere $\mathbb{S}(T)$ of $T$, if $T$ is equipped with a Euclidean norm, as in the proof of Lemma 8.9.

For a strict subset $A$ of $V$ of cardinality at least 2, define the diagonal $\Delta_A(\bar{S}_V(T))$ to be the subset of $\bar{S}_V(T)$ consisting of classes of maps $c$ from $V$ to $T$ that are constant on $A$ and such that $c(V \setminus A) \cap c(A) = \emptyset$. Let $\check{S}_V(T)$ denote the subspace of $\bar{S}_V(T)$ consisting of *injective* maps from $V$ to $T$ up to translation and dilation. The following theorem defines a bigger compactification of $\check{S}_V(T)$, which is also used in our study of the variations of $Z$, and especially in the definition of the anomalies. We will prove the following two theorems in Section 8.5. See Theorem 8.28 and Proposition 8.31.

**Theorem 8.11.** Set $S_{V,|V|}(T) = \bar{S}_V(T)$. For $k = |V| - 1, \dots, 3, 2$, define $S_{V,k}(T)$ from $S_{V,k+1}(T)$ by blowing up the closures of (the preimages under the composition of the previous blowdown maps of) the $\Delta_A(\bar{S}_V(T))$ such that $|A| = k$. At each step, the blown-up manifolds are smooth and transverse to the ridges, and the resulting blown-up manifold is independent of the order choice.

Thus, this process gives rise to a canonical compact smooth manifold $S_V(T) = S_{V,2}(T)$ with ridges. The interior of $S_V(T)$ is $\check{S}_V(T)$.

When $V$ has two elements, we have $S_V(T) = \check{S}_V(T) = \bar{S}_V(T)$. In general, the manifold $S_V(T)$ satisfies the following properties.
**Theorem 8.12.** With the notation of Theorem 8.11, for any subset A of V, the restriction from $\tilde{S}_V(T)$ to $\tilde{S}_A(T)$ extends to a smooth map from $S_V(T)$ to $S_A(T)$.

The open codimension-one faces of $S_V(T)$ will be the loci for which only one blow-up along some $\Delta_A(\overline{S}_V(T))$ is involved, for a strict subset A of V such that $|A| \ge 2$. The blowdown map sends such a face $f(A)(T)$ into $\Delta_A(\overline{S}_V(T))$. As it will be seen in Lemma 8.22, the fiber of the unit normal bundle of $\Delta_A(\overline{S}_V(T))$ is $\overline{S}_A(T)$. Thus, the following proposition will be clear in the end of Section 8.5, where a one-line proof is given.

**Proposition 8.13.** The codimension-one faces of $S_V(T)$ are in one-to-one correspondence with the strict subsets A of V with cardinality at least 2. The (open) face $f(A)(T)$ corresponding to such an A is

$$f(A)(T) = \tilde{S}_A(T) \times \tilde{S}_{\{a\} \cup (V \setminus A)}(T)$$

for an element a of A. For a subset e of cardinality 2 of V, the restriction to $f(A)(T)$ of the extended restriction

$$p_e: S_V(T) \to S_e(T)$$

may be described as follows:

* If $e \subseteq A$, then $p_e$ is the composition of the natural projections

$$f(A)(T) \to \tilde{S}_A(T) \to S_e(T).$$

* If $e \subseteq (V \setminus A) \cup \{a\}$, then $p_e$ is the composition of the natural projections

$$f(A)(T) \to \tilde{S}_{\{a\} \cup (V \setminus A)}(T) \to S_e(T).$$

* If $e \cap A = \{a'\}$, let $\tilde{e}$ be obtained from e by replacing $a'$ by a, then we have $p_e = p_{\tilde{e}}$.

The space $S_V(T_x \tilde{R})$ described in the above theorem is related to $C_V(R)$ by the following proposition, which is a corollary of Theorem 8.32.

**Proposition 8.14.** For $x \in \tilde{R}$, for $p_b: C_V(R) \to R^V$, we have

$$p_b^{-1}(x^V) = S_V(T_x \tilde{R}).$$
## 8.4 The codimension-one faces of $C(R, L; \Gamma)$

Recall that our terminology in Section 2.1.4 makes codimension-one faces open in the boundary of a manifold with ridges. Our codimension-one faces may be called *open codimension-one faces* in other references. We describe the codimension-one faces of the configuration space $C(R, L; \Gamma)$ for a link $L$ in a $\mathbb{Q}$-sphere $R$ below with an outline of justification. Details are provided in Sections 8.5 to 8.8. Again, the codimension-one faces are the loci where only one blow-up has been performed.

In this book, the sign $\subset$ stands for “is a strict subset of” or “$\subseteq$ and $\neq$”.

**Faces that involve blow-ups along diagonals $\Delta_A$.** Let $A$ be a subset of the set $V(\Gamma)$ of vertices of a Jacobi diagram $\Gamma$ on the domain $\mathcal{L}$ of a link $L$. Assume $|A| \ge 2$. Let us describe the (open) face $F(A, L, \Gamma)$, which comes from the blow-up along $\Delta_A(\check{R}^{V(\Gamma)})$. Such a face contains limit configurations that map $A$ to a point of $\check{R}$. An element of such a face is described by its image $m$ under the blowdown map

$$p_b: C_{V(\Gamma)}(R) \to R^{V(\Gamma)},$$

which maps $F(A, L, \Gamma)$ to $\Delta_A(\check{R}^{V(\Gamma)})$, together with an element of the fiber

$$\overline{\mathcal{S}}_A(T_{m(A)}\check{R}).$$

Let $a \in A$. Let $\check{\Delta}_A(\check{R}^{V(\Gamma)})$ denote the set of maps of $\Delta_A(\check{R}^{V(\Gamma)})$ whose restriction to $\{a\} \cup (V(\Gamma) \setminus A)$ is injective. Set

$$B(A, L, \Gamma) = \check{\Delta}_A(\check{R}^{V(\Gamma)}) \cap p_b(C(R, L; \Gamma)).$$

The face $F(A, L, \Gamma)$ fibers over the subspace $B(A, L, \Gamma)$. When $A$ contains no univalent vertex, the fiber over a point $m$ is $\check{\mathcal{S}}_A(T_{m(A)}\check{R})$.

Let $i_\Gamma$ be a $\Gamma$-compatible injection. Let $\mathcal{L}_1$ be an oriented connected component of $\mathcal{L}$. Let $U_1 = i_\Gamma^{-1}(\mathcal{L}_1)$. The restriction to $U_1$ of the injection $i_\Gamma$ into $\mathcal{L}_1$ induces a permutation $\sigma$ of $U_1$, such that, travelling along $\mathcal{L}_1$, we meet $i_\Gamma(v)$, $i_\Gamma(\sigma(v))$, ..., $i_\Gamma(\sigma^{|U_1|}(v) = v)$, successively, for any element $v$ of $U_1$. A set of consecutive elements of $U_1$, with respect to $i_\Gamma$, is a subset $A_U$ of $U_1$ that may be written as $\{v, \sigma(v), \dots, \sigma^k(v)\}$ for some element $v \in U_1$ and for $k \le |U_1| - 1$. If $A_U \ne U_1$, then the first element $v$ in such an $A_U$ is unique, and $\sigma$ induces the following unique linear order

$$v < \sigma(v) < \dots < \sigma^k(v)$$
on such a set $A_U$ of consecutive elements $U_1$, which is said to be compatible with the isotopy class $[i_\Gamma]$ of $i_\Gamma$. If $A_U = U_1$, every choice of an element $v$ in $A_U$ induces a linear (i.e., total) order

$$v < \sigma(v) < \dots < \sigma^k(v),$$

which is said to be compatible with $[i_\Gamma]$.

Let $A_U = A \cap U(\Gamma)$. When $A$ contains univalent vertices, if $F(A, L, \Gamma)$ is nonempty, then $A_U$ must be a set of consecutive vertices on a component $\mathcal{L}_1$ of $\mathcal{L}$ with respect to the given class $[i_\Gamma]$ of injections of $U(\Gamma)$ into $\mathcal{L}_1$. The fiber over a point $m$ is the subset $\tilde{S}_A(T_{m(A)}\tilde{R}, L, \Gamma)$ of $\tilde{S}_A(T_{m(A)}\tilde{R})$ consisting of injections that map $A_U$ on a line directed by $T_{m(A)}L$, so that the order induced by the line on $A_U$ coincides with

* the linear order induced by $[i_\Gamma]$, if $A_U$ is not the whole $i_\Gamma^{-1}(\mathcal{L}_1)$,

* one of the $|i_\Gamma^{-1}(\mathcal{L}_1)|$ linear orders compatible with the cyclic order induced by $[i_\Gamma]$, if $A_U = i_\Gamma^{-1}(\mathcal{L}_1)$.

In this latter case, neither the fiber nor $F(A, L, \Gamma)$ is connected. Their connected components are in one-to-one correspondence with the compatible orders.

According to Theorem 8.5, for any pair $e$ of $V(\Gamma)$, there exists a smooth restriction map from $C_{V(\Gamma)}$ to $C_e(R)$. An order on $e$ identifies $C_e(R)$ with $C_2(R)$. We describe the natural restriction $p_e$ to the (open) face $F(A, L, \Gamma)$ below for a pair $e$ of $V(\Gamma)$.

* If $|e \cap A| \le 1$, then $p_e$ is the composition of the natural projections

$$F(A, L, \Gamma) \to \check{\Delta}_A(\check{R}^{V(\Gamma)}) \to \check{C}_e(R).$$

* If $e \subseteq A$, then $p_e$ maps an element of $\tilde{S}_A(T_{m(A)}\tilde{R}, L, \Gamma)$ to its projection in $\tilde{S}_e(T_{m(A)}\tilde{R}) \subset C_e(R)$.

**Faces** $F(V(\Gamma), L, \Gamma)$. Let us study the following special case of the previous paragraph's faces, which will play a particular role. Consider a face $F(V(\Gamma), L, \Gamma)$ where a connected graph $\Gamma$ with at least one univalent vertex collapses and $L$ is a knot embedding. Such a face has one connected component for each linear order of $U(\Gamma)$ compatible with the cyclic order of $U(\Gamma)$.
**Notation 8.15.** A Jacobi diagram $\check{\Gamma}$ on $\mathbb{R}$ yields a diagram $c\ell(\check{\Gamma})$ on $S^1$, where $S^1$ is viewed as $\mathbb{R} \cup \{\infty\}$ by adding $\infty$ to $\mathbb{R}$. The natural orientation of $\mathbb{R}$ orders $U(\check{\Gamma})$. For a Jacobi diagram $\Gamma$ on $S^1$, represent a linear order of $U(\Gamma)$ compatible with the cyclic order of $U(\Gamma)$ by a Jacobi diagram $\check{\Gamma}$ on $\mathbb{R}$ such that $c\ell(\check{\Gamma}) = \Gamma$ and the order of $U(\Gamma)$ matches the order of $U(\check{\Gamma})$. Denote the corresponding connected component of $F(V(\Gamma), L, \Gamma)$ by $F(V(\Gamma), L, \check{\Gamma})$.

Such a connected component fibers over the domain $S^1$ of the knot. We denote the fiber over $z \in S^1$ by $\check{S}(T_{L(z)}\check{R}, \check{t}_{L(z)}; \check{\Gamma})$, where $\check{t}_{L(z)}$ is an oriented tangent vector to $L$ at $L(z)$.

The space $\check{S}(T_{L(z)}\check{R}, \check{t}_{L(z)}; \check{\Gamma})$ is the space of injections of $V(\check{\Gamma})$ into the vector space $T_{L(z)}\check{R}$ that map $U(\check{\Gamma})$ to the oriented line $\mathbb{R}\check{t}_{L(z)}$ directed by $\check{t}_{L(z)}$ with respect to the linear order of $U(\check{\Gamma})$, up to dilation and translation with respect to a vector in $\mathbb{R}\check{t}_{L(z)}$. It is naturally a subspace of $\check{S}_{V(\Gamma)}(T_{L(z)}\check{R})$. Indeed, moding out by all translations is equivalent to considering only configurations that map a univalent vertex to $\mathbb{R}\check{t}_{L(z)}$ and moding them out by translations along $\mathbb{R}\check{t}_{L(z)}$.

**Lemma 8.16.** The closure $\bar{F}(V(\Gamma), L, \check{\Gamma})$ of $F(V(\Gamma), L, \check{\Gamma})$ in $C(R, L; \Gamma)$ is a manifold with ridges. The closure $\mathcal{S}(T_{L(z)}\check{R}, \check{t}_{L(z)}; \check{\Gamma})$ of $\check{S}(T_{L(z)}\check{R}, \check{t}_{L(z)}; \check{\Gamma})$ in $C(R, L; \Gamma)$ is canonically diffeomorphic to its closure in $\mathcal{S}_{V(\Gamma)}(T_{L(z)}\check{R})$. It is a smooth manifold with ridges.

**PROOF:** The first assertion comes from Proposition 8.6 and from the fact that the closed faces of manifolds with ridges are manifolds with ridges (or from the proof of Lemma 8.17 at the end of Section 8.5). The space $\mathcal{S}(T_{L(z)}\check{R}, \check{t}_{L(z)}; \check{\Gamma})$ is the closure of $\check{S}(T_{L(z)}\check{R}, \check{t}_{L(z)}; \check{\Gamma})$ in the fiber over $L(z)^{V(\Gamma)}$ of $C_{V(\Gamma)}(\check{R})$, which is $\mathcal{S}_{V(\Gamma)}(T_{L(z)}\check{R})$ according to Proposition 8.14.

Now $\bar{F}(V(\Gamma), L, \check{\Gamma})$ fibers over $S^1$, and the fiber over $S^1$ is also a manifold with ridges. $\square$

Let $A$ be a strict subset of $V(\check{\Gamma})$ with cardinality at least 2 whose univalent vertices are consecutive on $\mathbb{R}$. Let $a \in A$. Let $\check{\Delta}_A(\mathcal{S}_{V(\Gamma)}(T_{L(z)}\check{R}))$ denote the set of (classes of) maps of $\mathcal{S}_{V(\Gamma)}(T_{L(z)}\check{R})$ whose restriction to $A$ is constant and whose restriction to $\{a\} \cup (V(\Gamma) \setminus A)$ is injective. Set

$$B(A, \vec{t}_{L(z)}; \check{\Gamma}) = \mathcal{S}(T_{L(z)}\check{R}, \vec{t}_{L(z)}; \check{\Gamma}) \cap \check{\Delta}_A(\mathcal{S}_{V(\Gamma)}(T_{L(z)}\check{R})).$$

Define the (open) face $f(A, \vec{t}_{L(z)}; \check{\Gamma})$ of $\mathcal{S}(T_{L(z)}\check{R}, \vec{t}_{L(z)}; \check{\Gamma})$ to be the space that fibers over the subspace $B(A, \vec{t}_{L(z)}; \check{\Gamma})$, whose fiber is the space of injections from $A$ into $T_{L(z)}\check{R}$ that map the univalent vertices of $A$ to the oriented line $\mathbb{R}\vec{t}_{L(z)}$ with respect to the linear order of $U(\check{\Gamma}) \cap A$, up to dilation and translation by a vector in $\mathbb{R}\vec{t}_{L(z)}$.

We will prove the following lemma at the end of Section 8.5.
**Lemma 8.17.** The codimension-one faces of $\mathcal{S}(T_{L(z)}\check{R}, \vec{t}_{L(z)}; \check{\Gamma})$ are the faces $f(A, \vec{t}_{L(z)}; \check{\Gamma})$ for the strict subsets $A$ of $V(\check{\Gamma})$ with cardinality at least 2 whose univalent vertices are consecutive on $\mathbb{R}$. The faces $f(A, \vec{t}_{L(z)}; \check{\Gamma})$ are the intersections of $\mathcal{S}(T_{L(z)}\check{R}, \vec{t}_{L(z)}; \check{\Gamma})$ with the codimension-one faces $f(A)(T_{L(z)}\check{R})$ of $\mathcal{S}_{V(\check{\Gamma})}(T_{L(z)}\check{R})$ listed in Proposition 8.13. In particular, the restriction maps $p_e$ from $f(A)(T_{L(z)}\check{R})$ to $\mathcal{S}_e(T_{L(z)}\check{R})$ of Proposition 8.13 restrict as restriction maps still denoted by $p_e$ from $f(A, \vec{t}_{L(z)}; \check{\Gamma})$ to $\mathcal{S}_e(T_{L(z)}\check{R})$.

**Faces that involve $\infty$.** Let $A$ be a nonempty subset of the set $V(\Gamma)$ of vertices of a Jacobi diagram $\Gamma$. Let $a \in A$. Let us describe the (open) face $F_\infty(A, L, \Gamma)$ that comes from the blow-up along $\Xi_A$. It contains limit configurations, which map $A$ to $\infty$. If it is nonempty, then $A$ contains no univalent vertices.

Let $\check{\Xi}_A$ denote the set of maps of $\Xi_A$ that restrict to injective maps on $\{a\} \cup (V(\Gamma) \setminus A)$, and set

$$B_\infty(A, L, \Gamma) = \check{\Xi}_A \cap p_b(C(R, L; \Gamma)).$$

The face $F_\infty(A, L, \Gamma)$ fibers over the subspace $B_\infty(A, L, \Gamma)$.

**Notation 8.18.** Let $\check{\mathcal{S}}(T_\infty R, A)$ denote the set of injective maps from $A$ to $(T_\infty R \setminus 0)$ up to dilation.

Note that $\check{\mathcal{S}}(T_\infty R, A)$ is an open submanifold of the unit normal bundle of $\Xi_A$, which is nothing but $\mathcal{S}((T_\infty R)^A)$. We have

$$F_\infty(A, L, \Gamma) = B_\infty(A, L, \Gamma) \times \check{\mathcal{S}}(T_\infty R, A).$$

For a pair $e$ of $V(\Gamma)$, the natural restriction to $F_\infty(A, L, \Gamma)$ of

$$p_e: C_{V(\Gamma)} \to C_e(R)$$

behaves in the following way.

* If $e \subseteq V(\Gamma) \setminus A$, then $p_e$ is the composition of the natural maps

$$F_\infty(A, L, \Gamma) \to B_\infty(A, L, \Gamma) \to \check{C}_{V(\Gamma)\setminus A}(R) \to C_e(R).$$

* If $e \subseteq A$, then $p_e$ is the composition of the natural maps

$$F_\infty(A, L, \Gamma) \to \check{\mathcal{S}}(T_\infty R, A) \to \check{\mathcal{S}}(T_\infty R, e) \hookrightarrow C_e(R).$$

* If $e \cap A = \{a'\}$, then $p_e$ is the composition of the natural maps

$$F_\infty(A, L, \Gamma) \to \check{C}_{e\setminus\{a'\}}(R) \times \check{\mathcal{S}}(T_\infty R, \{a'\}) \to \\ \to \check{R}^{e\setminus\{a'\}} \times \mathcal{S}(T_\infty R^{\{a'\}}) \hookrightarrow C_e(R).$$
**Summary.** We have just outlined a proof of the following proposition.

**Proposition 8.19.** *The disjoint union of*

* the faces $F_\infty(A, L, \Gamma)$ for nonempty subsets $A$ of $T(\Gamma)$ and

* the faces $F(A, L, \Gamma)$ for subsets $A$ of $V(\Gamma)$ with cardinality at least 2 such that $A \cap U(\Gamma)$ is a (possibly empty) set of consecutive vertices on a connected component of $\mathcal{L}$,

described above, embeds canonically in $C(R, L; \Gamma)$. Its image is the open codimension-one boundary $\partial_1 C(R, L; \Gamma) \setminus \partial_2 C(R, L; \Gamma)$ of $C(R, L; \Gamma)$. Furthermore, for any ordered pair of $V(\Gamma)$, the restriction to this codimension-one boundary of the map

$$p_e: C(R, L; \Gamma) \to C_2(R)$$

given by Theorem 8.4 is as described above.

Proposition 8.19 is proved in Section 8.8. It will be used to prove that $Z^s$ and $Z$ are independent of the used propagating forms of Theorems 7.20 and 7.40, in Chapters 9 and 10.

All the results stated so far in this chapter are sufficient to understand the proofs of Theorems 7.20 and 7.40. We prove them in detail in the rest of this chapter, which can be skipped by a reader already convinced by the outline above.

## 8.5 Detailed study of $S_V(T)$

In this section, we study the configuration space $S_V(T)$ presented in Section 8.3. We first prove Theorem 8.11. Let us first describe the transformations operated by the first blow-ups, locally.

Fix $T$, equip $T$ and its powers with Euclidean norms. Fix $\tilde{w}_0 \in \bar{S}_V(T)$ and $b(V) \in V$. Identify $\bar{S}_V(T)$ with the unit sphere $\mathbb{S}(T^{V\setminus\{b(V)\}})$ of $T^{V\setminus\{b(V)\}}$. Then $\tilde{w}_0$ is viewed as a map from $V$ to $T$ such that $\tilde{w}_0(b(V)) = 0$ and $\sum_{v \in V} \|\tilde{w}_0(v)\|^2 = 1$. Recall Definition 8.1 for a partition associated to a map and the associated notation.

**Definition 8.20.** A based partition of a finite set $V$ equipped with a basepoint $b(V) \in V$ is a partition $K(V)$ of $V$ into nonempty subsets $A$ equipped with basepoints $b(A)$ such that

* $b(A) \in A$, and
• if $b(V) \in A$, then $b(A) = b(V)$.

Fix $\tilde{w}_0$. Let $K(V) = K(V; \tilde{w}_0)$ be the associated fixed partition. Fix associated basepoints so that $K(V)$ becomes a based partition.

In general for a based subset $A$ of $V$ equipped with a based partition $(K(A), b)$, define the set $O(A, K(A), b, T)$ of maps $w: V \to T$ such that

• $\sum_{B \in K(A)} \|w(b(B))\|^2 = 1$,

• $w(b(A)) = 0$, $w(V \setminus A) = \{0\}$, and

• two elements of $A$ that belong to different kids of $A$ are mapped to different points of $T$ by $w$.

There is a straightforward identification of $O(V, K(V), b, T)$ with an open subset of $\overline{S}_V(T)$, which contains $\tilde{w}_0$. Let $w_0 \in O(V, K(V), b, T)$ denote the element that corresponds to $\tilde{w}_0$ under this identification. Set

$$W_V = O(V, K(V), b, T) \cap (\cap_{A \in D(V)} \Delta_A (\overline{S}_V(T))).$$

Note that $K(V)$ is a set that is naturally based by the element $b(K(V))$ of $K(V)$ that contains $b(V)$. Then $W_V$ is an open subset of $\mathbb{S}(T^{K(V)\setminus\{b(K(V))\}})$. It is the image $\tilde{\mathbb{S}}(T^{K(V)\setminus\{b(K(V))\}})$ of $\tilde{S}_{K(V)}(T)$ under the canonical identification of $\overline{S}_{K(V)}(T)$ with $\mathbb{S}(T^{K(V)\setminus\{b(K(V))\}})$.

For $A \in K(V)$, view the elements of $T^{A\setminus\{b(A)\}}$ as the maps from $V$ to $T$ that map $(V \setminus A) \cup \{b(A)\}$ to 0, and let $T_{<\varepsilon}^{A\setminus\{b(A)\}}$ denote the ball of its elements of norm smaller than $\varepsilon$. Note the easy lemma.

**Lemma 8.21.** Let $K(V)$ be a based partition of $V$. Let $w_0 \in W_V$. Then there exist an open neighborhood $N(w_0)$ of $w_0$ in $W_V$ and an $\varepsilon \in ]0, \infty[$ such that the map

$$N(w_0) \times \prod_{A \in D(V)} T_{<\varepsilon}^{A\setminus\{b(A)\}} \to \overline{S}_V(T) \\ (w, (\mu_A \tilde{w}_A)_{A \in D(V)}) \mapsto w + \sum_{A \in D(V)} \mu_A \tilde{w}_A$$

is an open embedding whose image does not meet diagonals that do not correspond to (nonstrict) subsets $B$ of daughters of $V$.

□

In particular, the first blow-ups that will affect this neighborhood of $\tilde{w}_0$ in $\overline{S}_V(T)$ are blow-ups along diagonals corresponding to daughters of $V$. For any daughter $A$ of $V$, the above identification identifies the normal bundle to $\Delta_A(\overline{S}_V(T))$ with $T^{A\setminus\{b(A)\}}$, and the corresponding blow-up replaces the factor $T_{<\varepsilon}^{A\setminus\{b(A)\}}$ with $[0, \varepsilon[ \times \mathbb{S}(T^{A\setminus\{b(A)\}})$. Thus, it is clear that the blow-ups corresponding to different daughters of $V$ commute. Note that our argument also proves the following lemma.
**Lemma 8.22.** Let A be a subset of V. The fiber of the unit normal bundle to $\Delta_A(\overline{\mathcal{S}}_V(T))$ is $\overline{\mathcal{S}}_A(T)$.

□

When performing the blow-ups along the diagonals corresponding to the daughters of V, we replace $\mu_A \tilde{w}_A \in T^{A\setminus\{b(A)\}}$, for $\mu_A \in [0, \varepsilon]$ and $\tilde{w}_A \in \mathbb{S}(T^{A\setminus\{b(A)\}})$, with $(\mu_A, \tilde{w}_A)$. Thus, we replace $0 \in \prod_{A \in D(V)} T^{A\setminus\{b(A)\}}$ with the set of normal vectors $\tilde{w}_A$ that pop up during the blow-ups.

**Lemma 8.23.** In particular, with the notation of Lemma 8.21, we get a chart of

$$ \mathcal{B}(O(V, K(V), b, T), (\Delta_A(\overline{\mathcal{S}}_V(T)) \cap O(V, K(V), b, T))_{A \in D(V)}), $$

which maps

$$ (w, (\mu_A, \tilde{w}_A)_{A \in D(V)}) \in N(w_0) \times \prod_{A \in D(V)} ([0, \varepsilon \times \mathbb{S}(T^{A\setminus\{b(A)\})]) $$

to the element

$$ (w + \sum_{A \in D(V)} \mu_A \tilde{w}_A, (\tilde{w}_A)_{A \in D(V): \mu_A=0}) $$

of

$$ \mathcal{B}(O(V, K(V), b, T), (\Delta_A(\overline{\mathcal{S}}_V(T)) \cap O(V, K(V), b, T))_{A \in D(V)}), $$

where the $\tilde{w}_A$ are the normal vectors viewed in $\overline{\mathcal{S}}_A(T)$ that popped up during the blow-ups.

We can construct an atlas of $\mathcal{B}(O(V, K(V), b, T), (\Delta_A(\overline{\mathcal{S}}_V(T))_{A \in D(V)})$ with charts of this form.

□

In order to conclude and get charts of manifolds with ridges, we blow up the

$$ \overline{\mathcal{S}}_A(T) \cong \mathbb{S}(T^{A\setminus\{b(A)\}}) $$

for the daughters A of V, and we iterate. Such an iteration produces a parenthesization of V as in Definition 8.2.

**Definition 8.24.** A Δ-parenthesization of V is a parenthesization P of V such that $V \in P$. The daughters of an element A of a parenthesization P (with respect to P) are the maximal elements of P strictly included in A.
The mother of an element $A$ in $\mathcal{P}$ is the smallest element of $\mathcal{P}$ that strictly contains $A$ (if there is one). A $\Delta$-parenthesization $\mathcal{P}$ is organized as a tree, in which the vertices are the elements of $\mathcal{P}$ and the edges are in one-to-one correspondence with the pairs (daughter, mother) of $\mathcal{P}^2$. We orient its edges from the daughter to her mother, as in Figure 8.3. The sons of an element $A$ of $\mathcal{P}$ are the singletons consisting of elements of $A$ that do not belong to a daughter of $A$ (with respect to $\mathcal{P}$). Any element $A$ of $\mathcal{P}$ is equipped with the set $K(A, \mathcal{P}) (= K(A))$ when $\mathcal{P}$ is fixed) of the kids of $A$, which are its daughters and its sons.

**Example 8.25.** The trees associated to the parenthesizations $\{V, A_{123}, A_{23}\}$ and $\{V, A_{12}, A_{34}\}$, which correspond to Figures 8.1 and 8.2 in Section 8.1, are pictured in Figure 8.3. With respect to the parenthesization $\{V, A_{123}, A_{23}\}$, the daughter $A_{123}$ of $V$ has two kids, its son $\{1\}$ and its daughter $A_{23}$.

<figure><img src="image_76.png" /><figcaption>Figure 8.3: Trees associated to $\Delta$-parenthesizations</figcaption></figure>

**Definition 8.26.** Let $\mathcal{T}(V)$ (resp. $\mathcal{T}_\Delta(V)$) denote the set of parenthesizations (resp. $\Delta$-parenthesizations) of $V$. Fix $\mathcal{P} \in \mathcal{T}_\Delta(V)$. Choose a basepoint $b(A) = b(A; \mathcal{P})$ for any $A \in \mathcal{P}$ so that if $A$ and $B$ are in $\mathcal{P}$, if $B \subset A$, and if $b(A) \in B$, then $b(B) = b(A)$. When $A \in \mathcal{P}$, $D(A)$ denotes the set of daughters of $A$. The basepoint $b(B)$ of a son $B$ of $A \in \mathcal{P}$ is its unique point. A $\Delta$-parenthesization equipped with basepoints as above is called a *based $\Delta$-parenthesization*.

Recall the canonical identification of the set $O(A, K(A), b, T)$ of Definition 8.20 with an open subset of $\bar{\mathcal{S}}_A(T)$. Set

$$W_A = O(A, K(A), b, T) \cap (\cap_{B \in D(A)} \Delta_B(\bar{\mathcal{S}}_A(T))).$$

Note that $W_A$ may be identified canonically with an open subset of the sphere $\bar{\mathcal{S}}_{K(A)}(T)$.

For $((\mu_A)_{A \in \mathcal{P}} \setminus \{V\}}, (w_A)_{A \in \mathcal{P}}) \in (\mathbb{R}^+)^{\mathcal{P} \setminus \{V\}} \times \prod_{A \in \mathcal{P}} W_A$, and for $B \in \mathcal{P}$, define $v_B((\mu_A)_{A \in \mathcal{P}} \setminus \{V\}}, (w_A)_{A \in \mathcal{P}})$ as the following map from $B$ to $T$:

$$ \begin{aligned} v_B((\mu_A), (w_A)) &= \sum_{C \in \mathcal{P}: C \subseteq B} \left( \prod_{D \in \mathcal{P}: C \subseteq D \subset B} \mu_D \right) w_C \\ &= w_B + \sum_{C \in D(B)} \mu_C \left( w_C + \sum_{D \in D(C)} \mu_D (w_D + \dots) \right). \end{aligned} $$
The construction of $v_B$ is illustrated in Figure 8.4. We can see it on the subtree whose vertices are the subsets of $B$. The map $v_B$ is the sum over the vertices $C$ of this subtree of the maps $w_C$ associated to its vertices, multiplied by the products of the coefficients $\mu_D$ associated to the edges of the path from $C$ to $B$.

<figure><img src="image_77.png" /><figcaption>Figure 8.4: About the construction of $v_B$</figcaption></figure>

Note that $v_B((\mu_A), (w_A)) \in O(B, K(B), b, T)$ when the $\mu_A$ are small enough. Also note the following easy lemma.

**Lemma 8.27.** For any $(w_A^0)_{A \in \mathcal{P}} \in \prod_{A \in \mathcal{P}} W_A$, there exists a neighborhood $N((w_A^0))$ of $0 \times (w_A^0)$ in $(\mathbb{R}^+)^{\mathcal{P}\setminus\{V\}} \times \prod_{A \in \mathcal{P}} W_A$ such that, for any $((\mu_A)_{A \in \mathcal{P}\setminus\{V\}}, (w_A)_{A \in \mathcal{P}}) \in N((w_A^0))$,

* if $v_V((\mu_A), (w_A))$ is constant on $B$ for a subset $B$ of $V$, then $B$ is included in (or equal to) a daughter of $V$,

* if $\mu_A \neq 0$ for any $A \in \mathcal{P}$, then $v_V((\mu_A), (w_A))$ is an injective map from $V$ to $T$.

□

When the construction of Theorem 8.11 makes sense, we denote a point of our configuration space $S_V(T)$ as a tuple

$$ \left( v_V((\mu_A), (w_A)), \left( v_B((\mu_A), (w_A)) \right)_{B: \mu_B=0} \right), $$

which contains its blowdown projection $v_V((\mu_A), (w_A))$ in $\bar{S}_V(T)$ followed by the normal vectors $v_B((\mu_A), (w_A))$ that have popped up during the blow-ups. Lemma 8.22 ensures that the normal vectors are elements of $\bar{S}_B(T)$ for some $B$. These normal vectors are nonconstant maps from $B$ to $T$ up to translation and dilation.

**Theorem 8.28.** Theorem 8.11 is correct and defines $S_V(T)$. Let $\mathcal{P}$ be a $\Delta$-parenthesization of $V$. Let $S_{V,\mathcal{P}}(T)$ be the space of the elements that have been transformed by the blow-ups along the closures of the $(\Delta_A(T))_{A \in \mathcal{P}\setminus\{V\}}$ and that have not been transformed by other blow-ups. Then $S_{V,\mathcal{P}}(T)$ is canonically diffeomorphic to

$$ \prod_{A \in \mathcal{P}} \check{S}_{K(A)}(T). $$
The composed blowdown map sends $((w_A)_{A \in \mathcal{P}}) \in \prod_{A \in \mathcal{P}} \check{S}_{K(A)}(T)$ to the map that sends an element of a kid $B$ of $V$ to $w_V(B)$. For $A \in \mathcal{P} \setminus \{V\}$, the $w_A$ are (similarly identified with) the unit normal vectors that have appeared during the blow-ups.

As a set, $S_V(T)$ is the disjoint union over the $\Delta$-parentheses $\mathcal{P}$ of $V$ of the $S_{V,\mathcal{P}}(T)$.

Any based $\Delta$-parenthesisization $\mathcal{P}$ of $V$ and any $(w_A^0)_{A \in \mathcal{P}} \in \prod_{A \in \mathcal{P}} W_A$ provide the following smooth open embedding $\psi(\mathcal{P}, (w_A^0)_{A \in \mathcal{P}})$ from a neighborhood $N((w_A^0))$ as in Lemma 8.27 to $S_V(T)$:

$$ N((w_A^0)) \quad \hookrightarrow \quad S_V(T) \\ ((\mu_A)_{A \in \mathcal{P}\setminus\{V\}}, (w_A)_{A \in \mathcal{P}}) \quad \mapsto \quad \left( \begin{array}{c} v_V((\mu_A), (w_A)), \\ (v_B((\mu_A), (w_A)))_{\{B \in \mathcal{P} \setminus \{V\} : \mu_B=0\}} \end{array} \right). $$

This embedding restricts to $N((w_A^0)) \cap ((\mathbb{R}^{+*})^{\mathcal{P}\setminus\{V\}} \times \prod_{A \in \mathcal{P}} W_A)$ as a diffeomorphism onto an open subset of $\check{S}_V(T)$. Furthermore, the open images of the embeddings $\psi(\mathcal{P}, (w_A^0)_{A \in \mathcal{P}})$ for different $(\mathcal{P}, (w_A^0)_{A \in \mathcal{P}})$ cover $S_V(T)$, and the codimension of $S_{V,\mathcal{P}}(T)$ in $S_V(T)$ equals the cardinality of $\mathcal{P} \setminus \{V\}$.

**PROOF:** Note that the images of the embeddings corresponding to $\mathcal{P} = \{V\}$ cover $\check{S}_V(T)$ trivially. The theorem is obviously true when $|V| = 2$. We proceed by induction on $|V|$. By induction, for each daughter $A$ of $V$, the space $S_A(T)$ is covered by images of smooth open embeddings $\psi(\mathcal{P}_A, (w_B^0)_{B \in \mathcal{P}_A})$ from $N((w_B^0)_{B \in \mathcal{P}_A})$ to some open subset $U_A$ of $S_A(T)$ associated to parentheses $\mathcal{P}_A$ of $A$, as in the statement. So Lemma 8.23 leaves us with the proof that $\psi(\mathcal{P}, (w_A^0)_{A \in \mathcal{P}})$ is smooth and open for any $(w_A^0)_{A \in \mathcal{P}} \in \prod_{A \in \mathcal{P}} W_A$, for a small neighborhood $N((w_A^0))$ satisfying the conditions of Lemma 8.27. It suffices to check that the change of coordinates replacing the coordinates $(\tilde{\mu}_A, \tilde{w}_A)$ written as $(\mu_A, \tilde{w}_A)$ in Lemma 8.23 by the coordinates $(\mu_A, v_A)$ compatible with $\psi(\mathcal{P}, (w_A^0)_{A \in \mathcal{P}})$ is smooth and open for each daughter $A$ of $V$. We have $\tilde{\mu}_A \tilde{w}_A = \mu_A v_A$, $\sum_{a \in A} \|\tilde{w}_A(a)\|^2 = 1$, and $\sum_{B \in D(A)} \|v_A(b(B))\|^2 = 1$. We get

$$ \tilde{w}_A = \left( \sum_{a \in A} \|v_A(a)\|^2 \right)^{-1/2} v_A, \quad v_A = \left( \sum_{B \in D(A)} \|\tilde{w}_A(b(B))\|^2 \right)^{-1/2} \tilde{w}_A, $$

$$ \tilde{\mu}_A = \left( \sum_{a \in A} \|v_A(a)\|^2 \right)^{1/2} \mu_A, \text{ and } \mu_A = \left( \sum_{B \in D(A)} \|\tilde{w}_A(b(B))\|^2 \right)^{1/2} \tilde{\mu}_A, $$
where $\sum_{a \in A} \|v_A(a)\|^2 \ge 1$, and $\sum_{B \in D(A)} \|\tilde{w}_A(b(B))\|^2 > 0$. $\square$

The above proof also proves the following two propositions, with the notation $\partial_r$ introduced in the beginning of Section 2.1.4.

**Proposition 8.29.** Let $\mathcal{T}_{r,\Delta}(V)$ be the subset of $\mathcal{T}_{\Delta}(V)$ consisting of the $\Delta$-parenthesized sets of $V$ of cardinality $r$. Then

$$ \partial_{r-1}(S_V(T)) \setminus \partial_r(S_V(T)) = \sqcup_{P \in \mathcal{T}_{r,\Delta}(V)} S_{V,P}(T). $$

For two $\Delta$-parenthesized sets $P$ and $P'$, if $P \subset P'$, then $S_{V,P'}(T) \subset \overline{S_{V,P}(T)}$.

**PROOF:** We can deduce the first assertion from the charts of Theorem 8.28. Let $c_0 = (w_A)_{A \in \mathcal{P}'} \in S_{V,P'}(T)$ and let $\Psi = \psi(\mathcal{P}', (w_A)_{A \in \mathcal{P}'}) : N((w_A)) \to S_V(T)$ be a smooth open embedding as in Theorem 8.28. Let $\varepsilon \in ]0, \infty[$ be such that $[0, \varepsilon]^{\mathcal{P}'\setminus\{V\}} \times \{(w_A)_{A \in \mathcal{P}'}\} \subset N((w_A)_{A \in \mathcal{P}'})$. For $t \in [0, 1]$, set

$$ \mu_A(t) = \begin{cases} \varepsilon t & \text{if } A \in \mathcal{P}' \setminus \mathcal{P} \\ 0 & \text{if } A \in \mathcal{P} \setminus \{V\} \end{cases} $$

and let $c(t) = \Psi((\mu_A(t))_{A \in \mathcal{P}'\setminus\{V\}}, (w_A)_{A \in \mathcal{P}'})$. Then $c(t) \in S_{V,P}(T)$ for any $t \in ]0, 1]$, and $\lim_{t \to 0} c(t) = c_0$. $\square$

**Proposition 8.30.** Any injective linear map $\phi$ from a vector space $T$ to another such $T'$ induces a canonical embedding $\phi_* : S_V(T) \to S_V(T')$. This embedding maps an element $((w_A)_{A \in \mathcal{P}})$ of $S_{V,P}(T)$ to the element $((\phi \circ w_A)_{A \in \mathcal{P}})$ of $S_{V,P}(T')$. If $\psi$ is another injective linear map from a vector space $T'$ to a third vector space $T''$, then we have $(\psi \circ \phi)_* = \psi_* \circ \phi_*$. $\square$

Finally, let us check the following proposition, which implies Theorem 8.12.

**Proposition 8.31.** Let $A$ be a finite subset of cardinality at least 2 of a finite set $V$. For $P \in \mathcal{T}_{\Delta}(V)$, define

$$ P_A = \{B \cap A : B \in P, |B \cap A| \ge 2\} $$

in $\mathcal{T}_{\Delta}(A)$. For $C \in P_A$, let $\hat{C}$ be the smallest element of $P$ that contains $C$ or is equal to $C$. For

$$ w = (w_B \in \check{S}_{K(B)}(T))_{B \in \mathcal{P}} \in S_{V,P}(T), $$

and for $C \in P_A$, define $w'_C$ to be the natural restriction of $w_{\hat{C}}$ to $K(C, P_A)$. Then set

$$ p_A(w) = ((w'_C)_{C \in \mathcal{P}_A}) \in S_{A,P_A}(T). $$
This consistently defines a smooth map

$$p_A: S_V(T) \to S_A(T).$$

The map $p_A$ is the unique continuous extension from $S_V(T)$ to $S_A(T)$ of the restriction map from $\tilde{S}_V(T)$ to $\tilde{S}_A(T)$.

PROOF: It is easy to see that this restriction map is well-defined. In order to prove that it is smooth, use the charts of Theorem 8.28. Fix a $\Delta$-parenthesization $\mathcal{P}$ of $V$. Let $\mathcal{P}_A$ be the induced $\Delta$-parenthesization of $A$. Fix basepoints $b_A$ for $\mathcal{P}_A$ according to the rule in Definition 8.26. Fix basepoints for the elements $B$ of $\mathcal{P}$ so that $b(B) = b_A(A \cap B)$ when $B \cap A \in \mathcal{P}_A$, and $b(B) \in B \cap A$ when $B \cap A \neq \emptyset$. According to Theorem 8.28, it suffices to prove smoothness in charts involving the based $\Delta$-parenthesizations $\mathcal{P}$ and $\mathcal{P}_A$. So it suffices to prove that the projections on the factors that contain the restrictions maps $w'_C$ are smooth for $C \in \mathcal{P}_A$, and that the projections on the factors that contain the dilation factors $\mu_C$ are smooth for $C \in \mathcal{P}_A \setminus \{A\}$.

With our charts and our conditions on the basepoints, for any $C \in \mathcal{P}_A$, we have

$$w'_C = \frac{1}{g(C, w_{\hat{C}})} w_{\hat{C}}|_{K(C, \mathcal{P}_A)},$$

where

$$g(C, w_{\hat{C}}) = \sqrt{\sum_{D \in K(C, \mathcal{P}_A)} \|w_{\hat{C}}(D)\|^2}$$

is not zero since $w_{\hat{C}}$ is nonconstant on $K(C, \mathcal{P}_A)$. So, the projection on the factor of $w'_C$ is smooth.

Let $C \in \mathcal{P}_A \setminus \{A\}$, and let $m(C)$ denote the mother of $C$ in $\mathcal{P}_A$. Let $E$ be the set of basepoints of the kids of $m(C)$ distinct from $C$ with respect to $\mathcal{P}_A$. Consider the elements $B_i \in \mathcal{P}$, for $i = 1, 2, \dots, k(B)$, such that $C = B_i \cap A$ for any $i \le k(C)$, where $(\hat{C} = B_1) \subset B_2 \cdots \subset B_{k(C)}$. We have $b(B_i) = b_A(C)$ for any $i \in k(C)$ and $b(\widehat{m(C)}) = b_A(m(C))$.

Then the restrictions of the configurations $w'_{m(C)} + \mu_C w'_C$ and

$$w_{\widehat{m(C)}} + \left( \prod_{i=1}^{k(C)} \mu_{B_i} \right) w_{\hat{C}}$$

to $C \cup E$ coincide up to dilation. So $g(m(C), w_{\widehat{m(C)}}) \mu_C w'_C$ coincides with

$$\left( \prod_{i=1}^{k(C)} \mu_{B_i} \right) g(C, w_{\hat{C}}) w'_C$$
on $C$, and we have

$$ \mu_C = \left( \prod_{i=1}^{k(C)} \mu_{B_i} \right) \frac{g(C, w_{\hat{C}})}{g(m(C), w_{\overleftarrow{m(C)}})}. $$

Thus $\mu_C$ is smooth (it is defined even when the $\mu_{B_i}$ are negative). $\square$

Proposition 8.13 follows from Propositions 8.29 and 8.31. $\square$

PROOF OF LEMMA 8.17: The structure of a smooth manifold with ridges of $S(T_{L(z)}\check{R}, \check{t}_{L(z)}; \check{\Gamma})$ in Lemma 8.16 can be alternatively deduced from the charts of Theorem 8.28. These charts also show that $S(T_{L(z)}\check{R}, \check{t}_{L(z)}; \check{\Gamma})$ is a submanifold transverse to the ridges of $S_{V(\check{\Gamma})}(T_{L(z)}\check{R})$. So its codimension-one faces are the intersections of $S(T_{L(z)}\check{R}, \check{t}_{L(z)}; \check{\Gamma})$ with the codimension-one faces of $S_{V(\check{\Gamma})}(T_{L(z)}\check{R})$. Then Lemma 8.17 follows from Proposition 8.13. $\square$

## 8.6 Blowing up diagonals

In the rest of this chapter, $M$ is a smooth manifold without boundary of dimension $\delta > 0$. It is not necessarily oriented. The set of injective maps from $V$ to $M$ is denoted by $\check{C}_V[M]$ with brackets instead of parentheses (we have $\check{C}_V[\check{R}] = \check{C}_V(\check{R})$, but $\check{C}_V[R] \neq \check{C}_V(R)$).

**Theorem 8.32.** Let $M$ be a manifold. Let $V$ be a finite set.

Set $C_{V,|V|+1}[M] = M^V$. For $k = |V|, \dots, 3, 2$, define $C_{V,k}[M]$ from $C_{V,k+1}[M]$ by blowing up the closures of (the preimages under the composition of the previous blowdown maps of) the $\Delta_A(M^V)$ such that $|A| = k$. At each step, the blown-up manifolds are smooth and transverse to the ridges, and the resulting blown-up manifold is independent of the order choice. Thus, this process gives rise to a canonical smooth manifold $C_V[M] = C_{V,2}[M]$ with ridges equipped with its composed blowdown projection

$$ p_b: C_V[M] \to M^V. $$

* Let $f \in M^V$ be a map from $V$ to $M$. Then $p_b^{-1}(f)$ is canonically diffeomorphic to $\prod_{A \in D(V;f)} S_A(T_{f(A)}M)$. We will denote an element $x \in p_b^{-1}(f)$ of $C_V[M]$ by $(f \in M^V, (w_A \in S_A(T_{f(A)}M))_{A \in D(V;f)})$, with the notation of Definition 8.1.

* For any open subset $U$ of $M$, $C_V[U] = p_b^{-1}(U^V)$.

* The space $\check{C}_V[M]$ is dense in $C_V[M]$.
• If $M$ is compact, then $C_V[M]$ is compact, too.

• Any choice of a basepoint $b = b(V)$ of $V$ and of an open embedding

$$ \phi: \mathbb{R}^{\delta} \to M $$

induces the diffeomorphism

$$ \psi(\phi, b) : \mathbb{R}^{\delta} \times \mathbb{R}^{+} \times \mathcal{S}_{V}(\mathbb{R}^{\delta}) \to C_{V}[\phi(\mathbb{R}^{\delta})] $$

described below.

Let $u \in \mathbb{R}^{\delta}$, $\mu \in \mathbb{R}^{+}$, and $n \in \mathcal{S}_{V}(\mathbb{R}^{\delta})$. Then $\psi(\phi, b)$ satisfies

$$ p_b(\psi(\phi, b)(u, \mu, n)) = \phi \circ (u + \mu p_b(n)), $$

where $p_b(n)$ is viewed as a map from $V$ to $\mathbb{R}^{\delta}$ such that $p_b(n)(b) = 0$ and $\sum_{v \in V} \|p_b(n)(v)\|^2 = 1$, and $(u + \mu p_b(n))$ denotes the map from $V$ to $\mathbb{R}^{\delta}$ obtained from $p_b(n)$ by composition by the homothety with ratio $\mu$, followed by the translation of vector $u$. For $u \in \mathbb{R}^{\delta}$ and $n \in \tilde{\mathcal{S}}_{V}(\mathbb{R}^{\delta})$, the map $p_b(\psi(\phi, b)(u, 0, n))$ is constant with value $\phi(u)$ and we have

$$ \psi(\phi, b)(u, 0, n) = (\phi(u)^{V}, (T_{u}\phi)_{*}(n)). $$

The restriction of $\psi(\phi, b)$ to $\mathbb{R}^{\delta} \times \mathbb{R}^{+*} \times \tilde{\mathcal{S}}_{V}(\mathbb{R}^{\delta})$ is a diffeomorphism onto $\tilde{C}_{V}[\phi(\mathbb{R}^{\delta})]$.

• An embedding $\phi$ of a manifold $M_1$ into another such $M_2$ induces the canonical embedding $\phi_*$ such that

$$ \phi_* (f \in M_1^V, (w_A \in \mathcal{S}_A(T_{f(A)}M_1))_{A \in D(V;f)}) = (\phi \circ f, (T_{f(A)}\phi)_*(w_A)) $$

from $C_V[M_1]$ to $C_V[M_2]$. If $\psi$ is an embedding from $M_2$ to another manifold $M_3$, then we have $(\psi \circ \phi)_* = \psi_* \circ \phi_*$.

PROOF: Start with $M = \mathbb{R}^{\delta}$ equipped with its usual Euclidean norm. Fix $b(V)$. Any map $f$ from $V$ to $\mathbb{R}^{\delta}$ may be written as $f(b(V)) + y$, for a unique element $y \in (\mathbb{R}^{\delta})^{V \setminus \{b(V)\}}$. Then blowing up $(\mathbb{R}^{\delta})^V$ along the diagonal $\Delta_V((\mathbb{R}^{\delta})^V)$, described by the equation $y = 0$, replaces $(\mathbb{R}^{\delta})^{V \setminus \{b(V)\}}$ with $\mathbb{R}^+ \times \mathcal{S}((\mathbb{R}^{\delta})^{V \setminus \{b(V)\}})$ and provides a diffeomorphism from $\mathbb{R}^{\delta} \times \mathbb{R}^+ \times \mathcal{S}((\mathbb{R}^{\delta})^{V \setminus \{b(V)\}})$ to $\mathcal{B}((\mathbb{R}^{\delta})^V, \Delta_V((\mathbb{R}^{\delta})^V))$.

The diagonals corresponding to strict subsets of $V$ are products by $\mathbb{R}^{\delta} \times \mathbb{R}^{+}$ of the diagonals corresponding to the same subsets for the manifold $\mathcal{S}((\mathbb{R}^{\delta})^{V \setminus \{b(V)\}}) \cong \bar{\mathcal{S}}_V(\mathbb{R}^{\delta})$, which was studied in the previous subsection.
Thus, $C_V[\mathbb{R}^\delta]$ is well-defined, and our diffeomorphism lifts to a diffeomorphism from $\mathbb{R}^\delta \times \mathbb{R}^+ \times S_V(\mathbb{R}^\delta)$ to $C_V[\mathbb{R}^\delta]$. So the composition of this diffeomorphism with the product of the charts obtained in Theorem 8.28 by the identity map yields an atlas of $C_V[\mathbb{R}^\delta]$.

For a diffeomorphism $\phi$ from $\mathbb{R}^\delta$ to an open subspace $U$ of a manifold, the diffeomorphism

$$ \phi^V: (\mathbb{R}^\delta)^V \to U^V $$

preserves diagonals. So $C_V[U]$ is well-defined for any open subset $U$ diffeomorphic to $\mathbb{R}^\delta$, in a manifold $M$. Furthermore, $\phi^V$ lifts as a natural diffeomorphism $\phi_*: C_V[\mathbb{R}^\delta] \to C_V[U]$.

Note that the elements of $C_V[\mathbb{R}^\delta]$ have the prescribed form. Since the normal bundle to a diagonal $\Delta_A(\tilde{C}_V[U])$ is $((TU^A/\Delta_A(TU))\setminus\{0\})/\mathbb{R}^{+*}$, the diffeomorphism $\phi_*$ from $C_V[\mathbb{R}^\delta]$ to $C_V[U]$ maps

$$ x = (f \in (\mathbb{R}^\delta)^V, (w_A \in S_A(\mathbb{R}^\delta))_{A \in D(V;f)}) $$

to

$$ \phi_*(x) = (\phi \circ f, ((T_{f(A)}\phi)_*(w_A))_{A \in D(V;f)}). $$

Then the elements of $C_V[U]$ have the prescribed form, too.

In order to prove that $C_V[M]$ is well-defined for a manifold $M$, it suffices to see that it is well-defined over an open neighborhood of any point $c$ of $M^V$. Such a map $c$ defines the partition $K(V; c)$. There exist pairwise disjoint open neighborhoods $U_A$ diffeomorphic to $\mathbb{R}^\delta$ of the $c(A)$, for $A \in K(V) = K(V; c)$. It is easy to see that $C_V[M]$ is well-defined over $\prod_{A \in K(V)} U_A^A$ and that it is canonically isomorphic to $\prod_{A \in K(V)} C_A[U_A]$, there. Thus, $C_V[M]$ is well-defined, and its elements have the prescribed form.

If $\phi$ is a diffeomorphism from a manifold $M_1$ to another such $M_2$, then the diffeomorphism

$$ \phi^V: M_1^V \to M_2^V $$

preserves diagonals. So it lifts as a natural diffeomorphism $\phi_*: C_V[M_1] \to C_V[M_2]$, which behaves as stated.

Then the study of the map induced by an embedding from a manifold $M_1$ into another such $M_2$ can be easily reduced to the case of a linear embedding from $\mathbb{R}^k$ to $\mathbb{R}^\delta$. For such an embedding, use the identification of $C_V[\mathbb{R}^\delta]$ with $\mathbb{R}^\delta \times \mathbb{R}^+ \times S_V(\mathbb{R}^\delta)$.

The other statements are easy to check.

□

**Proposition 8.33.** Let $B$ be a subset of $V$, then the restriction from $\tilde{C}_V[M]$ to $\tilde{C}_B[M]$ extends uniquely to a smooth map $p_B$ from $C_V[M]$ to $C_B[M]$. Let
$f \in M^V$. The elements $A$ of $D(B; f|_B)$ are of the form $B \cap \hat{A}$ for a unique $\hat{A}$ in $D(V; f)$. Then we have

$$p_B(f, (w_C \in S_C(T_{f(C)}M))_{C \in D(V;f)}) = (f|_B, (w_{\hat{A}}|_A)_{A \in D(B;f|_B)}).$$

**PROOF:** It is obvious when $|B| = 1$. Assume that $|B| \ge 2$. In order to check that this restriction map is smooth, it is enough to prove that it is smooth when $M = \mathbb{R}^\delta$. Assume that $b(V) \in B$. Use the diffeomorphism of Theorem 8.32 to write an element of $C_V[\mathbb{R}^\delta]$ as $(u \in \mathbb{R}^\delta, \mu \in \mathbb{R}^+, n \in S_V(\mathbb{R}^\delta))$. Then the restriction maps $(u, \mu, n)$ to $(u, \|p_b(n)|_B\|\mu, n|_B)$. It is smooth according to Proposition 8.31. $\square$

An element $x = (f \in M^V, (w_A \in S_A(T_{f(A)}M))_{A \in D(V;f)})$ of $C_V[M]$ induces the parenthesization $\mathcal{P}(x)$ of $V$ that is the union over the elements $A$ of $D(V; f)$ of the $\mathcal{P}_A(x)$ such that $w_A \in S_{A, \mathcal{P}_A(x)}(T_{f(A)}M)$. (See Theorem 8.28.) Let $\mathcal{P}$ be a parenthesization of $V$. Set $C_{V, \mathcal{P}}[M] = \{x \in C_V[M] : \mathcal{P}(x) = \mathcal{P}\}$.

The following proposition is easy to observe. (See Proposition 8.29.)

**Proposition 8.34.** Let $\mathcal{T}_r(V)$ be the subset of $\mathcal{T}(V)$ consisting of the parenthensizations of $V$ of cardinality $r$. Then we have

$$\partial_r(C_V[M]) \setminus \partial_{r+1}(C_V[M]) = \sqcup_{\mathcal{P} \in \mathcal{T}_r(V)} C_{V, \mathcal{P}}[M].$$

Furthermore, for any two parenthensizations $\mathcal{P}$ and $\mathcal{P}'$ such that $\mathcal{P} \subset \mathcal{P}'$, we have $C_{V, \mathcal{P}'}[M] \subset \overline{C_{V, \mathcal{P}}[M]}$.

$\square$

## 8.7 Blowing up $\infty$

For a finite set $V$, let $V^+ = V \cup \{v_\infty\}$. We have $V^+ = V \cup \{v_\infty\}$.

We state and prove the following generalization of Theorem 8.4, where $C_V(R) = C_V[R, \infty]$.

**Theorem 8.35.** Let $V$ be a finite set. Let $M$ be a manifold without boundary of dimension $\delta$, and let $\infty \in M$. Set $\tilde{M} = M \setminus \{\infty\}$. Recall the manifold $C_{V^+}[M]$ of Theorem 8.32. Let $p_b|_{\{v_\infty\}}: C_{V^+}[M] \to M$ map a configuration to its value at $v_\infty$. Define $C_V^+[M, \infty]$ to be the preimage of $\infty$ under this map. Then $C_V^+[M, \infty]$ is a smooth submanifold of $C_{V^+}[M]$ transverse to the ridges equipped with charts induced by the local models of Theorems 8.32 and 8.28.
It is the closure in $C_{V^+}[M]$ of $\check{C}_V[\tilde{M}] \times \{\infty\}^{\{v_\infty\}}$, where $\check{C}_V[\tilde{M}] \subset \tilde{M}^V$ is the space of injective maps from $V$ to $\tilde{M}$.

Furthermore, the manifold $C_V^+[M, \infty]$ is canonically diffeomorphic to the manifold $C_V[M, \infty]$ obtained from $M^V$ by the following process.

For a nonempty $A \subseteq V$, let $\Xi_A$ be the set of maps from $V$ to $M$ that map $A$ to $\infty$, and $V \setminus A$ to $\tilde{M}$.

Start with $M^V$. Blow up $\Xi_V$, which is reduced to the point $m = \infty^V$ such that $m^{-1}(\infty) = V$. Set

$$C_{V,|V|+1}[M, \infty] = B\ell(M^V, \infty^V).$$

Then for $k = |V|, |V| - 1, \dots, 3, 2$, define $C_{V,k}[M, \infty]$ from $C_{V,k+1}[M, \infty]$ by blowing up the closures of (the preimages under the composition of the previous blowdown maps of) the $\Delta_A(\tilde{M}^V)$ such that $|A| = k$ and the closures of (the preimages under the composition of the previous blowdown maps of) the $\Xi_J$ such that $|J| = k-1$. At each step, the blown-up manifolds are smooth and transverse to the ridges, and $C_{V,k}[M, \infty]$ is independent of the possible order choice of the blow-ups. The obtained manifold $C_V[M, \infty] = C_{V,2}[M, \infty]$ is a smooth manifold of dimension $\delta|V|$, with ridges. It is compact if $M$ is compact. The interior of $C_V[M, \infty]$ is $\check{C}_V[\tilde{M}]$, and the composition of the blowdown maps gives rise to a canonical smooth blowdown projection $p_b: C_V[M, \infty] \to M^V$.

**PROOF:** Let $U_\infty$ be a small open neighborhood of $\infty$ in $M$. Then the local models given by a composition of the local models of Theorems 8.32 and 8.28 make clear that $C_V^+[U_\infty, \infty]$ has a canonical smooth structure and that $C_V^+[U_\infty, \infty]$ is the closure in $C_{V^+}[U_\infty]$ of $\check{C}_V[U_\infty \setminus \{\infty\}] \times \{\infty\}^{\{v_\infty\}}$.

Let $U$ be an open subset of $M$ disjoint from $U_\infty$. Let $A$ be a subset of $V$. Consider the map $p_b|_V: C_V^+[M, \infty] \to M^V$. Observe

$$(p_b|_V)^{-1}(U_\infty^{A^+} \times U^{V\setminus A}) = C_A^+[U_\infty, \infty] \times C_{V\setminus A}[U].$$

The preimage of $(U_\infty^A \times U^{V\setminus A})$ under $p_b: C_V[M, \infty] \to M^V$ is similarly canonically diffeomorphic to $C_A[U_\infty, \infty] \times C_{V\setminus A}[U]$.

So we are left with the identification of a small open neighborhood of $(p_b|_V)^{-1}(\infty^V)$ in $C_V^+[U_\infty, \infty]$ and a small open neighborhood of $p_b^{-1}(\infty^V)$ in $C_V[U_\infty, \infty]$ for an arbitrarily small neighborhood $U_\infty$ of $\infty$ and a finite set $V$. Assume without loss that $U_\infty$ is identified with $\mathbb{R}^\delta$ by a diffeomorphism $\varphi: \mathbb{R}^\delta \to U_\infty$ such that $\varphi(0) = \infty$. The naturality of the constructions leaves us with the case $(U_\infty, \infty) = (\mathbb{R}^\delta, 0)$.
Equip $V^+$ with the basepoint $v_\infty$. The first blow-up along the small diagonal of $V^+$ transforms $(\mathbb{R}^\delta)^{V^+}$ to

$$C_{V^+,|V^+|}[\mathbb{R}^\delta] \cong (\mathbb{R}^\delta)^{\{v_\infty\}} \times \mathbb{R}^+ \times \bar{S}_{V^+}(\mathbb{R}^\delta),$$

where the space of $\bar{S}_{V^+}(\mathbb{R}^\delta)$ of nonconstant maps from $V^+$ to $\mathbb{R}^\delta$ up to translation and dilation is naturally identified with the space $S((\mathbb{R}^\delta)^V)$ of nonzero maps from $V$ to $\mathbb{R}^\delta$ up to dilation. The first factor $(\mathbb{R}^\delta)^{\{v_\infty\}}$ contains the value of the configuration at $v_\infty$. It is zero on $p_b(C_V^+[\mathbb{R}^\delta, 0])$. Since $S((\mathbb{R}^\delta)^V)$ is the unit normal bundle of $\{0\}^V$ in $M^V$, we get a canonical diffeomorphism from $C_{V^+,|V^+|}[\mathbb{R}^\delta] \cap (p_b|_{\{v_\infty\}})^{-1}(0)$ to $C_{V,|V|+1}[\mathbb{R}^\delta, 0]$.

Proceed by induction to get a canonical diffeomorphism from $C_{V^+,k}[\mathbb{R}^\delta] \cap (p_b|_{\{v_\infty\}})^{-1}(0)$ to $C_{V,k}[\mathbb{R}^\delta, 0]$ for all $k$. $\square$

Theorem 8.5 is a direct consequence of Proposition 8.33 and Theorem 8.35. (Recall $C_V(R) = C_V[R, \infty]$.) $\square$

An element $x$ of $C_V^+[M, \infty]$ induces a parenthesization $\mathcal{P}^+(x) = \mathcal{P}(x \in C_{V^+}[M])$ of $V^+$ as before Proposition 8.34. By Theorems 8.28 and 8.32 the element $x$ can be written as

$$(p_b(x)|_V, (w_B \in \check{S}_{K(B)}(T_{x(B)}M))_{B \in \mathcal{P}^+(x)}).$$

Let $\mathcal{P}_s^+(x)$ denote the set of elements of $\mathcal{P}^+(x)$ containing $v_\infty$. This set is totally ordered by the inclusion. So is

$$\mathcal{P}_s(x) = \{A \setminus \{v_\infty\} : A \in \mathcal{P}_s^+(x)\} = \{V(1), V(2), \dots, V(\sigma)\}$$

with $V(i+1) \subset V(i)$. We have $x(V(i)) = \infty$ for all $i$. Let $K_d^s(V(i))$ denote the set of kids of $V(i)^+$ that do not contain $v_\infty$.

Recall from Notation 8.18 that $\check{S}(T_\infty M, A)$ denotes the set of injective maps from $A$ to $(T_\infty M \setminus 0)$ up to dilation. The natural basepoint choice of $v_\infty$ identifies the set $\check{S}_{K(V(i)^+)}(T_\infty M)$ of injective maps from $K(V(i)^+)$ to $T_\infty M$ up to dilation and translation with $\check{S}(T_\infty M, K_d^s(V(i)))$.

Set $\mathcal{P}_d(x) = \mathcal{P}^+(x) \setminus \mathcal{P}_s^+(x)$. When $x$ is seen as an element of $C_V[M, \infty]$, the set $\mathcal{P}_s(x)$ is the set of subsets $B$ of $V$ such that $x$ has been transformed by the blow-up along (the closure of the preimage of) $\Xi_B$ and $\mathcal{P}_d(x)$ is the set of subsets $B$ of $V$ such that $x$ has been transformed by the blow-up along (the closure of the preimage of) $\Delta_B(M^V)$.

The proof of Theorem 8.35 also proves the following proposition.

**Proposition 8.36.** Let $\mathcal{P}^+$ be a parenthesization of $V^+ = V \sqcup \{v_\infty\}$. Set

$$C_{V,\mathcal{P}^+}[M, \infty] = \{x \in C_V^+[M, \infty] : \mathcal{P}^+(x) = \mathcal{P}^+\}.$$
Use the canonical identification of $C_V^+[M, \infty]$ with $C_V[M, \infty]$. Then the stratum $C_{V,\mathcal{P}^+}[M, \infty]$ is an open part of

$$ \partial_{|\mathcal{P}^+|}(C_V[M, \infty]) \setminus \partial_{|\mathcal{P}^+|+1}(C_V[M, \infty]). $$

Let $\mathcal{P}_s^+$ be the set of elements of $\mathcal{P}^+$ containing $v_\infty$. Set $\mathcal{P}_d = \mathcal{P}^+ \setminus \mathcal{P}_s^+$ and $\mathcal{P}_s = \{A \setminus \{v_\infty\} : A \in \mathcal{P}_s^+\}$. For an element $B$ of $\mathcal{P}_s \cup \{V\}$, let $K_d^s(B)$ denote the set of kids of $B^+$ that do not contain $v_\infty$. The stratum $C_{V,\mathcal{P}^+}[M, \infty]$ fibers over the space $\check{K}_d^s(V)[\check{M}]$ of injective maps $c$ from $K_d^s(V)$ to $\check{M}$. Its fiber over such an injective map $c$ is

$$ \left( \prod_{B \in \mathcal{P}_s} \check{S}(T_\infty M, K_d^s(B)) \right) \times \left( \prod_{A \in \mathcal{P}_d} \check{S}_{K(A)}(T_{c(A)} M) \right) = \prod_{A \in \mathcal{P}^+} \check{S}_{K(A)}(T_{c(A)} M). $$

**Corollary 8.37.** An element of $C_V[M, \infty]$ is a map $c$ from $V$ to $M$, equipped with

* a parenthesization $\mathcal{P}^+$ of $V^+ = V \sqcup \{v_\infty\}$ and induced parenthesizations $\mathcal{P}_s$ and $\mathcal{P}_d$ of $V$, as in Proposition 8.36,

* an element $f_B$ of $\check{S}(T_\infty M, K_d^s(B))$ for any element $B$ of $\mathcal{P}_s$,

* an element $w_A$ of $\check{S}_{K(A)}(T_{c(A)} M)$ for each $A \in \mathcal{P}_d$.

## 8.8 Finishing the proofs of the statements of Sections 8.2 and 8.4

**PROOF OF PROPOSITION 8.6:** To study the closure of $\check{C}(R, L; \Gamma)$ in $C_{V(\Gamma)}(R)$, we study its intersection with some $p_b^{-1}(\prod_{i \in I} U_i^{V_i})$ for disjoint small compact $U_i$. We assume that at most one $U_i$ contains $\infty$ and that this $U_i$ does not meet the link. So the corresponding $V_i$ does not contain univalent vertices, and Theorem 8.4 gives the structure of the corresponding factor.

Thus, it is enough to study $p_A(\check{C}(R, L; \Gamma)) \cap C_A(\phi(\mathbb{R}^3))$ when

* $\phi$ is an embedding from $\mathbb{R}^3$ to $\check{R}$ that maps the vertical line $\mathbb{R}\vec{v}$ through the origin oriented from bottom to top onto $\phi(\mathbb{R}^3) \cap L$, so that $\phi$ identifies $(\mathbb{R}^3, \mathbb{R}\vec{v})$ with $(\phi(\mathbb{R}^3), \phi(\mathbb{R}^3) \cap L)^4$,

* the univalent vertices of $A$ form a nonempty subset $A_U = U(\Gamma) \cap A$ of consecutive vertices on the component $L_i$ of $L$ such that $\phi(\mathbb{R}^3) \cap L = \phi(\mathbb{R}^3) \cap L_i$.

<footnote n="4">⁴Here, $L$ also denotes the image $L(\mathcal{L})$ of $L$.</footnote>
Let $\mathcal{O}(A_U)$ denote the set of the linear orders $<$ on $A_U$ compatible with $[i_\Gamma]$.
(This is a singleton unless $A_U$ contains all the univalent vertices of $L_i$.) Via
the natural maps induced by $\phi$, the space $\check{C}_A(\phi(\mathbb{R}^3))$ is identified with the
set of injections from $A$ to $\mathbb{R}^3$, and $p_A(\check{C}(R, L; \Gamma)) \cap \check{C}_A(\phi(\mathbb{R}^3))$ is identified
with the disjoint union over $\mathcal{O}(A_U)$ of the subsets $\check{C}_A(\mathbb{R}^3, A_U, < \in \mathcal{O}(A_U))$ of
injections that map $A_U$ to $\mathbb{R}\vec{v}$ so that the order induced by $\mathbb{R}\vec{v}$ coincides with
$<$. Fix $< \in \mathcal{O}(A_U)$, and write $A_U = \{v_1, \dots, v_k\}$ so that $v_1 < v_2 < \dots < v_k$.

Fix $b(A) = v_1$, and study the closure of $\check{C}_A(\mathbb{R}^3, A_U, <)$ in $\mathbb{R}^3 \times \mathbb{R}^+ \times \mathcal{S}_A(\mathbb{R}^3)$, using the diffeomorphism $\psi(\phi, v_1)$ of Theorem 8.32. This closure is the closure of $\mathbb{R}\vec{v} \times \mathbb{R}^+ \times \check{\mathcal{S}}_A(\mathbb{R}^3, A_U, <)$, where $\check{\mathcal{S}}_A(\mathbb{R}^3, A_U, <)$ is the quotient of $\check{C}_A(\mathbb{R}^3, A_U, <)$ by translations by vectors of $\mathbb{R}\vec{v}$ and dilations. Then the charts of Theorem 8.28 (used with basepoints that are as much as possible in $A_U$) make clear that the closure of $\check{\mathcal{S}}_A(\mathbb{R}^3, A_U, <)$ in $\mathcal{S}_A(\mathbb{R}^3)$ consists of the limit configurations $c$ such that $(c(v_j) - c(v_i))$ is nonnegatively colinear with $\vec{v}$ at any scale (i.e., in any infinitesimal configuration $w$ that has popped up during the blow-ups) for any $i$ and $j$ in $\{1, \dots, k\}$ such that $i < j$, and that this closure is a smooth submanifold of $\mathcal{S}_A(\mathbb{R}^3)$ transverse to the ridges. $\square$

This proof also proves the following lemma.

**Lemma 8.38.** The codimension-one faces of $C(R, L; \Gamma)$ are the intersections of $C(R, L; \Gamma)$ with the codimension-one faces of $C_{V(\Gamma)}(R)$.

<figure><img src="image_78.png" /></figure>

Proposition 8.19 then follows, with the help of Propositions 8.33 and 8.36.

<figure><img src="image_79.png" /></figure>

**8.9 Alternative descriptions of configuration spaces**

Apart from Lemma 8.39, this section will not be used in this book. It men-
tions other presentations of the configuration spaces studied in Sections 8.5
to 8.7 without proofs. Most of the proofs are left to the reader as exercises.

Let $V$ be a finite set of cardinality at least 2. Let $\mathcal{P}_{\ge 2} = \mathcal{P}_{\ge 2}(V)$ be the set of its (nonstrict) subsets of cardinality at least 2. The smooth blowdown projection from $\mathcal{S}_A(T)$ to $\bar{\mathcal{S}}_A(T)$ for an $A \in \mathcal{P}_{\ge 2}(V)$ may be composed with the smooth restriction map from $\mathcal{S}_V(T)$ to $\mathcal{S}_A(T)$ to produce a smooth map $\pi_A$ from $\mathcal{S}_V(T)$ to $\bar{\mathcal{S}}_A(T)$.
**Lemma 8.39.** The product over the subsets A of V with cardinality at least 2 of the $\pi_A$ is a smooth embedding

$$ \prod_{A \in \mathcal{P}_{\ge 2}} \pi_A: S_V(T) \hookrightarrow \prod_{A \in \mathcal{P}_{\ge 2}} \bar{S}_A(T) $$

and the image of $S_V(T)$ is the closure of the image of the restriction of $\prod_{A \in \mathcal{P}_{\ge 2}} \pi_A$ to $\tilde{S}_V(T)$.

**PROOF:** The injectivity of $\prod_{A \in \mathcal{P}_{\ge 2}} \pi_A$ can be seen from the description of $S_V(T)$ as a set, which is given in Theorem 8.28. $\square$

**Proposition 8.40.** The image $(\prod_{A \in \mathcal{P}_{\ge 2}} \pi_A)(S_V(T))$ is the subset of

$$ \prod_{A \in \mathcal{P}_{\ge 2}} \bar{S}_A(T) $$

consisting of the elements $((c_A)_{A \in \mathcal{P}_{\ge 2}})$ such that for any two elements A and B of $\mathcal{P}_{\ge 2}$ such that $B \subset A$, the restriction of $c_A$ to B coincides with $c_B$ if it is not constant.

**PROOF:** Exercise. $\square$

Thus, $S_V(T)$ can be defined as its image described in the above proposition. Similar definitions involving only cardinality 2 or 3 subsets of V can be found in [Sin04].

For a smooth manifold M without boundary, we have similar smooth maps $\pi_A$ from $C_V[M]$ to $\mathcal{B}(M^A, \Delta_A(M^A))$. They also define a smooth map

$$ \prod_{A \in \mathcal{P}_{\ge 2}} \pi_A: C_V[M] \hookrightarrow \prod_{A \in \mathcal{P}_{\ge 2}} \mathcal{B}(M^A, \Delta_A(M^A)). $$

The elements of $\mathcal{B}(M^A, \Delta_A(M^A))$ are maps $c$ from A to M that are equipped with an element $w \in \bar{S}_A(T_{c(A)}M)$ when they are constant.

**Proposition 8.41.** The map

$$ \prod_{A \in \mathcal{P}_{\ge 2}} \pi_A: C_V[M] \hookrightarrow \prod_{A \in \mathcal{P}_{\ge 2}} \mathcal{B}(M^A, \Delta_A(M^A)) $$

is an embedding. Its image is the subset of $\prod_{A \in \mathcal{P}_{\ge 2}} \mathcal{B}(M^A, \Delta_A(M^A))$ consisting of the elements $((c_A)_{A \in \mathcal{P}_{\ge 2}})$ such that for any two elements A and B of $\mathcal{P}_{\ge 2}$ such that $B \subset A$,
• the restriction to $B$ of the map $p_b(c_A): A \to M$ coincides with $p_b(c_B)$, and,

• if $p_b(c_A)$ is constant, then the restriction to $B$ of $(w_A(c_A) \in \bar{S}_A(T_{c_A(A)}M))$ is $w_B(c_B)$ if this restriction is not constant.

PROOF: Exercise. □

Again, $C_V[M]$ can be defined as its image described in the above proposition, and similar definitions involving only cardinality 2 and 3 subsets of $V$ can be found in [Sin04].

We may obtain similar statements for $C_V[M, \infty]$, where $\infty \in M$, from the definition of $C_V[M, \infty]$ as $C_V^+[M, \infty]$ in Theorem 8.35.

More information about the homotopy groups and the homology of the configuration spaces $\tilde{C}_V[\mathbb{R}^d]$ and $\tilde{C}_V[S^d]$ can be found in the book [FH01] by Edward Fadell and Sufian Husseini. See also the book [CILW24] by Ricardo Campos, Najib Idrissi, Pascal Lambrechts, and Thomas Willwacher.
# Chapter 9

## Dependence on the propagating forms

This chapter shows how our combinations of integrals over configuration spaces depend on the chosen propagating forms.

### 9.1 Introduction

In this section, we give a first general description of the variation of $Z$ when propagating forms change in Proposition 9.2. Then we show how this proposition and a preliminary lemma (9.1) apply to prove Theorem 7.19 and two other lemmas (9.5 and 9.6), about independence of chosen propagating forms as in Definition 3.11. The typical proof of Proposition 9.2 will occupy the next sections.

Again, any closed 2-form on $\partial C_2(R)$ extends to $C_2(R)$ because the restriction induces a surjective map $H^2(C_2(R); \mathbb{R}) \to H^2(\partial C_2(R); \mathbb{R})$ since we have

$$H^3(C_2(R), \partial C_2(R); \mathbb{R}) = 0.$$

**Lemma 9.1.** Let $(\tilde{R}, \tau_0)$ be an asymptotic rational homology $\mathbb{R}^3$ as in Definition 3.8. Let $\tau: [0, 1] \times \tilde{R} \times \mathbb{R}^3 \to T\tilde{R}$ be a smooth map whose restriction to $\{t\} \times \tilde{R} \times \mathbb{R}^3$ is an asymptotically standard parallelization $\tau_t$ of $\tilde{R}$ for any $t \in [0, 1]$. Define $p_\tau: [0, 1] \times \partial C_2(R) \to [0, 1] \times S^2$ by $p_\tau(t, x) = (t, p_{\tau_t}(x))$.

Let $\omega_0$ and $\omega_1$ be two propagating forms of $C_2(R)$ that restrict to $\partial C_2(R) \setminus UB_R$ as $p_{\tau_0}^*(\omega_{0,S^2})$ and as $p_{\tau_1}^*(\omega_{1,S^2})$, respectively, for two volume-one forms $\omega_{0,S^2}$ and $\omega_{1,S^2}$ of $S^2$.

Then there exist

* a closed 2-form $\tilde{\omega}_{S^2}$ on $[0, 1] \times S^2$ whose restriction to $\{t\} \times S^2$ is $\omega_{t,S^2}$ for $t \in \{0, 1\}$,
* for any such $\tilde{\omega}_{S^2}$, a closed 2-form $\omega^\partial$ on $[0, 1] \times \partial C_2(R)$ whose restriction to $\{t\} \times \partial C_2(R)$ is $\omega_t|_{\partial C_2(R)}$ for $t \in \{0, 1\}$, and whose restriction to $[0, 1] \times (\partial C_2(R) \setminus U(B_R))$ is $p_\tau^*(\tilde{\omega}_{S^2})$, and,

* for any such compatible $\tilde{\omega}_{S^2}$ and $\omega^\partial$, a closed 2-form $\omega$ on $[0, 1] \times C_2(R)$ whose restriction to $\{t\} \times C_2(R)$ is $\omega_t$ for $t \in \{0, 1\}$, and whose restriction to $[0, 1] \times \partial C_2(R)$ is $\omega^\partial$.

If $\omega_0$ and $\omega_1$ are propagating forms of $(C_2(R), \tau_0)$ and $(C_2(R), \tau_1)$, then we may choose $\omega^\partial = p_\tau^*(\tilde{\omega}_{S^2})$ on $[0, 1] \times \partial C_2(R)$.

PROOF: As in Lemma 3.17, there exists a one-form $\eta_{S^2}$ on $S^2$ such that $d\eta_{S^2} = \omega_{1,S^2} - \omega_{0,S^2}$. Define the closed 2-form $\tilde{\omega}_{S^2}$ on $[0, 1] \times S^2$ by

$$ \tilde{\omega}_{S^2} = p_{S^2}^*(\omega_{0,S^2}) + d.tp_{S^2}^*(\eta_{S^2}), $$

where $t$ is the coordinate on $[0, 1]$.

Now, the form $\omega^\partial$ is defined on the boundary of $[0, 1] \times U(B_R)$, and it extends as a closed 2-form $\omega^\partial$ as desired there because the restriction induces a surjective map $H^2([0, 1] \times U(B_R); \mathbb{R}) \to H^2(\partial([0, 1] \times U(B_R)); \mathbb{R})$ since we have

$$ H^3([0, 1] \times U(B_R), \partial([0, 1] \times U(B_R)); \mathbb{R}) = 0. $$

Finally, the desired form $\omega$ is defined on the boundary of $[0, 1] \times C_2(R)$. It similarly extends as a closed 2-form to $[0, 1] \times C_2(R)$. $\square$

When $A$ is a subset of the set of vertices $V(\Gamma)$ of a numbered Jacobi diagram $\Gamma$ with support a one-manifold $\mathcal{L}$, the set of edges of $\Gamma$ between two elements of $A$ is denoted by $E(\Gamma_A)$ (edges of $\Gamma$ are plain), and $\Gamma_A$ is the subgraph of $\Gamma$ consisting of the vertices of $A$ and the edges of $E(\Gamma_A)$ together with the natural restriction to $U(\Gamma) \cap A$ of the isotopy class of injections from $U(\Gamma)$ to $\mathcal{L}$ associated to $\Gamma$.

The following proposition, whose proof occupies most of this chapter, is crucial in the study of the variations of $Z$.

**Proposition 9.2.** Let $(\tilde{R}, \tau)$ be an asymptotic rational homology $\mathbb{R}^3$. Let $L = \bigsqcup_{j=1}^k K_j$ be an embedding of $\mathcal{L} = \bigsqcup_{j=1}^k S_j^1$ into $\tilde{R}$. Let $\tau: [0, 1] \times \tilde{R} \times \mathbb{R}^3 \to T\tilde{R}$ be a smooth map whose restriction to $\{t\} \times \tilde{R} \times \mathbb{R}^3$ is an asymptotically standard parallelization $\tau_t$ of $\tilde{R}$ for any $t \in [0, 1]$.<fnref n="1" /> Define $p_\tau: [0, 1] \times \partial C_2(R) \to [0, 1] \times S^2$ by $p_\tau(t, x) = (t, p_{\tau_t}(x))$.

Let $n$ and $m$ be positive integers. For $i \in \underline{m}$, let $\tilde{\omega}(i)$ be a closed 2-form on $[0, 1] \times C_2(R)$ whose restriction to $\{t\} \times C_2(R)$ is denoted by $\tilde{\omega}(i, t)$, for any $t \in [0, 1]$.

<footnote n="1">¹This homotopy $\tau$ is not useful for this statement, but we will use this notation later.</footnote>
Assume that $\tilde{\omega}(i)$ restricts to $[0, 1] \times (\partial C_2(R) \setminus UB_R)$ as $p_\tau^*(\tilde{\omega}_{S^2}(i))$, for some closed two-form $\tilde{\omega}_{S^2}(i)$ on $[0, 1] \times S^2$ such that $\int_{\{t\} \times S^2} \tilde{\omega}_{S^2}(i) = 1$ for $t \in [0, 1]$. Recall Notation 7.29 and set

$$Z_{n,m}(t) = \sum_{\Gamma \in \mathcal{D}_{n,m}^e(\mathcal{L})} \zeta_{\Gamma} I(R, L, \Gamma, (\tilde{\omega}(i,t))_{i \in \Gamma}) [\Gamma] \in \mathcal{A}_n(\bigsqcup_{j=1}^k S_j^1)$$

and $Z_n(t) = Z_{n,3n}(t)$. Let $\mathcal{D}_{n,m}^{e,F}(\mathcal{L})$ denote the set of pairs $(\Gamma, A)$ such that

* $\Gamma \in \mathcal{D}_{n,m}^e(\mathcal{L})$,

* $A \subseteq V(\Gamma)$, $|A| \ge 2$,

* $\Gamma_A$ is a connected component of $\Gamma$,

* $|A| \equiv 2 \mod 4$ if $A \cap U(\Gamma) = \emptyset$, and

* $A \cap U(\Gamma)$ is a set of consecutive vertices on a component $\mathcal{L}_A$ of $\mathcal{L}$ if $A \cap U(\Gamma) \neq \emptyset$.

For $(\Gamma, A) \in \mathcal{D}_{n,m}^{e,F}(\mathcal{L})$, set

$$I(\Gamma, A) = \int_{[0,1] \times F(A,L,\Gamma)} \bigwedge_{e \in E(\Gamma)} p_e^*(\tilde{\omega}(j_E(e))) [\Gamma],$$

where $p_e: [0, 1] \times C(R, L; \Gamma) \to [0, 1] \times C_2(R)$ is the product by the identity map $\mathbf{1}_{[0,1]}$ of $[0, 1]$ of the previous $p_e$, $j_E$ is the edge-numbering map of Definition 7.6, and the face $F(A, L, \Gamma)$ of $C(R, L; \Gamma)$ is described in Section 8.4. Set $\mathcal{D}_n^{e,F}(\mathcal{L}) = \mathcal{D}_{n,3n}(\mathcal{L})$. Then we have

$$Z_n(1) - Z_n(0) = \sum_{(\Gamma,A) \in \mathcal{D}_n^{e,F}(\mathcal{L})} \zeta_{\Gamma} I(\Gamma, A)$$

and

$$Z_{n,3n-2}(1) - Z_{n,3n-2}(0) = \sum_{(\Gamma,A) \in \mathcal{D}_{n,3n-2}^{e,F}(\mathcal{L})} \zeta_{\Gamma} I(\Gamma, A).$$

This statement simplifies as in Corollary 9.4 when $L = \emptyset$ using the projection $p^c: \mathcal{A}(\emptyset) \to \mathcal{A}^c(\emptyset)$, which maps diagrams with several connected components to 0. Recall that $\mathcal{D}_n^c$ is the subset of $\mathcal{D}_n^e(\emptyset)$ whose elements are the numbered diagrams of $\mathcal{D}_n^e(\emptyset)$ with one connected component.

For an oriented connected trivalent diagram $\Gamma$, the face $F(V(\Gamma), \emptyset, \Gamma)$ fibers over $\tilde{R}$, and the fiber over $x \in \tilde{R}$ is the space $\tilde{S}_{V(\Gamma)}(T_x \tilde{R})$ of injections from $V(\Gamma)$ to $T_x \tilde{R}$, up to translation and dilation. See Section 8.3. We also denote this face by $\tilde{S}_{V(\Gamma)}(T \tilde{R})$.
**Lemma 9.3.** Let $\Gamma \in \mathcal{D}_n^c$ be equipped with a vertex-orientation, which induces an orientation of $C(R, \emptyset; \Gamma)$ as in Corollary 7.2. These orientations induce the orientation of $V(\Gamma)$ described in Remark 7.5. The orientation of $F(V(\Gamma), \emptyset, \Gamma)$ as part of the boundary of $C(R, \emptyset; \Gamma)$ can be alternatively described as follows. The face $F(V(\Gamma), \emptyset, \Gamma)$ is oriented as the local product $\tilde{R} \times \text{fiber}$, where the fiber is oriented as in Convention 8.10, using the above orientation of $V(\Gamma)$.

**PROOF:** The dilation factor for the quotient $\tilde{S}_{V(\Gamma)}(T_x \tilde{R})$ plays the role of an inward normal for $C(R, \emptyset; \Gamma)$. The orientation of $C(R, \emptyset; \Gamma)$ near the boundary is given by the orientation of $\tilde{R}$, followed by this inward normal, followed by the fiber orientation. $\square$

For any pair $e$ of $V(\Gamma)$, we have a natural restriction map

$$p_e: \tilde{S}_{V(\Gamma)}(T\tilde{R}) \to \tilde{S}_e(T\tilde{R}) \cong U\tilde{R},$$

which provides natural restriction maps

$$p_e: [0, 1] \times \tilde{S}_{V(\Gamma)}(T\tilde{R}) \to [0, 1] \times \tilde{S}_e(T\tilde{R})$$

by multiplication by $\mathbf{1}_{[0,1]}$.

Proposition 9.2 has the following corollary.

**Corollary 9.4.** Assume $L = \emptyset$. Recall Notation 7.16. Under the assumptions of Proposition 9.2, set $z_n(t) = p^c(Z_n(t))$ so that we have

$$z_n(t) = z_n\left(R, (\tilde{\omega}(i, t))_{i \in \underline{3n}}\right).$$

Then $z_n(1) = z_n(0)$ if $n$ is even, and $(z_n(1) - z_n(0))$ depends only on $(\tilde{\omega}(i)|_{[0,1] \times UB_R})_{i \in \underline{3n}}$ if $n$ is odd. Define

$$z_n([0, 1] \times UB_R; (\tilde{\omega}(i))_{i \in \underline{3n}}) = \sum_{\Gamma \in \mathcal{D}_n^c} \zeta_{\Gamma} I(\Gamma, V(\Gamma)),$$

where

$$I(\Gamma, V(\Gamma)) = \int_{[0,1] \times \tilde{S}_{V(\Gamma)}(TB_R)} \bigwedge_{e \in E(\Gamma)} p_e^*(\tilde{\omega}(j_E(e))) [\Gamma].$$

Then we have

$$z_n(1) - z_n(0) = z_n([0, 1] \times UB_R; (\tilde{\omega}(i))_{i \in \underline{3n}})$$

for any odd integer $n$.
PROOF: We have

$$ \int_{[0,1] \times \check{S}_{V(\Gamma)}(T(\check{R} \setminus B_R))} \bigwedge_{e \in E(\Gamma)} p_e^*(\tilde{\omega}(j_E(e))) = 0 $$

because the integrated form factors through $[0, 1] \times \check{S}_{V(\Gamma)}(\mathbb{R}^3)$ via a map induced by $p_\tau$, which is fixed and independent of $\tau$, there. In particular, the $I(\Gamma, V(\Gamma))$ depend only on $(\tilde{\omega}(i)|_{[0,1] \times U_{B_R}})_{i \in \underline{3n}}$. $\square$

PROOF OF THEOREM 7.19 ASSUMING PROPOSITION 9.2: Changing propagating forms $\omega(i)_0$ of $C_2(R)$ to other ones $\omega(i)_1$ provides forms $\tilde{\omega}(i)$ on $[0, 1] \times C_2(R)$ as in Lemma 9.1. Then Corollary 9.4 guarantees that $z_{2n}(\check{R}, \emptyset, (\omega(i)))$ does not depend on the used propagating forms (which are not normalized on $U(B_R)$ and hence do not depend on parallelizations). $\square$

**Lemma 9.5.** Let $(\check{R}, \tau)$ be an asymptotic rational homology $\mathbb{R}^3$. Let $L: \mathcal{L} \hookrightarrow \check{R}$ be a link embedding. For any $i \in \underline{3n}$, let $\omega(i)$ be a homogeneous propagating form of $(C_2(R), \tau)$. Then, as stated in Theorem 7.20, $Z_n(\check{R}, L, (\omega(i)))$ is independent of the chosen $\omega(i)$.<fnref n="2" /> Denote it by $Z_n(\check{R}, L, \tau)$.

PROOF ASSUMING PROPOSITION 9.2: To prove this lemma, it suffices to prove that if some homogeneous $\omega(i) = \tilde{\omega}(i, 0)$ is changed to another homogeneous propagating form $\tilde{\omega}(i, 1)$, then $Z_n(\check{R}, L, (\omega(i)))$ is unchanged. According to Lemma 3.17, under these assumptions, there exists a one-form $\eta$ on $C_2(R)$ such that

* $\tilde{\omega}(i, 1) = \tilde{\omega}(i, 0) + d\eta$ and

* $\eta|_{\partial C_2(R)} = 0$.

Let $p_{C_2}: [0, 1] \times C_2(R) \to C_2(R)$ denote the projection on the second factor. Define closed 2-forms $\tilde{\omega}(j)$ on $[0, 1] \times C_2(R)$ by

* $\tilde{\omega}(j) = p_{C_2}^*(\omega(j))$ if $j \neq i$, and

* $\tilde{\omega}(i) = p_{C_2}^*(\tilde{\omega}(i, 0)) + d.tp_{C_2}^*(\eta)$.

Then the variation of $Z_n(\check{R}, L, (\omega(i)))$ is $Z_n(1) - Z_n(0)$, with the notation of Proposition 9.2, where all the forms involved in some $I(\Gamma, A)$, except $p_{e(i)}^*(\tilde{\omega}(i))$, for the possible edge $e(i)$ such that $j_E(e(i)) = i$, factor through $p_{C_2}^*$. Thus, if $i \notin \text{Im}(j_E)$, then all the forms factor through $p_{C_2}^*$, and $I(\Gamma, A)$ vanishes.

<footnote n="2">²The proof of Theorem 7.20 will be concluded in the end of Section 10.5.</footnote>
Locally, $F(A, L, \Gamma)$ is diffeomorphic to the product of $F(A, L, \Gamma_A)$ by $C(R, L; \Gamma \setminus \Gamma_A)$. If $e(i)$ is not an edge of $\Gamma_A$, then the form $\bigwedge_{e \in E(\Gamma_A)} p_e^*(\tilde{\omega}(j_E(e)))$ factors through $F(A, L, \Gamma_A)$ whose dimension is $2|E(\Gamma_A)| - 1$, and the form vanishes. If $e(i)$ is an edge of $\Gamma_A$, then the part $p_{e(i)}^*(d.tp_{C_2}^*(\eta))$ vanishes since $\eta$ vanishes on $\partial C_2(R)$. Thus $\bigwedge_{e \in E(\Gamma_A)} p_e^*(\omega(j_E(e)))$ still factors through $F(A, L, \Gamma_A)$. So, when $(R, L, \tau)$ is fixed, $Z_n(\tilde{R}, L, (\omega(i)))$ is independent of the chosen homogeneous $\omega(i)$. $\square$

**Lemma 9.6.** Let $(\tilde{R}, \tau_0)$ be an asymptotic rational homology $\mathbb{R}^3$. Let $n \in \mathbb{N}$. For any $i \in \underline{3n}$, let $\omega(i)$ be a propagating form of $(C_2(R), \tau_0)$. Then $Z_n(\tilde{R}, \emptyset, (\omega(i))) = Z_n(\tilde{R}, \emptyset, \tau_0)$ with the notation of Lemma 9.5. Furthermore, $Z_n(\tilde{R}, \emptyset, \tau_0)$ depends only on the homotopy class of $\tau_0$.

**PROOF ASSUMING PROPOSITION 9.2:** Let $\tau: [0, 1] \times \tilde{R} \times \mathbb{R}^3 \to T\tilde{R}$ be a smooth map whose restriction to $\{t\} \times \tilde{R} \times \mathbb{R}^3$ is an asymptotically standard parallelization $\tau_t$ of $\tilde{R}$ for any $t \in [0, 1]$. Define $p_\tau: [0, 1] \times \partial C_2(R) \to [0, 1] \times S^2$ by $p_\tau(t, x) = (t, p_{\tau_t}(x))$. For any $i \in \underline{3n}$, let $\omega(i)_0$ be a (non-necessarily homogeneous) propagating form of $(C_2(R), \tau_0)$, and let $\omega(i)_1$ be a propagating form of $(C_2(R), \tau_1)$. It suffices to prove that

$$Z_n(\tilde{R}, \emptyset, (\omega(i)_1)) - Z_n(\tilde{R}, \emptyset, (\omega(i)_0)) = 0.$$

Use forms $\omega(i)$ on $[0, 1] \times C_2(R)$ provided by Lemma 9.1, which restrict to $[0, 1] \times \partial C_2(R)$ as $p_\tau^*(\tilde{\omega}_{S^2})$, to express this variation as in Proposition 9.2. Here, a face $F(A, \emptyset, \Gamma)$ is an open dense subset of the product of $F(A, \emptyset, \Gamma_A)$ by $\tilde{C}(R, \emptyset; \Gamma \setminus \Gamma_A)$, and $\tau$ identifies $[0, 1] \times F(A, \emptyset, \Gamma_A)$ with $[0, 1] \times \tilde{R} \times \tilde{S}_A(\mathbb{R}^3)$. The form $\bigwedge_{e \in E(\Gamma)} p_e^*(\omega(j_E(e)))$ pulls back through $[0, 1] \times \tilde{S}_A(\mathbb{R}^3) \times \tilde{C}(R, \emptyset; \Gamma \setminus \Gamma_A)$, and it vanishes. $\square$

The following variant of Proposition 9.2 implies Theorem 7.30. We prove it in Section 9.3.

**Proposition 9.7.** Under the assumptions of Proposition 9.2, the following statement is also true. For $\Gamma \in \mathcal{D}_{n,m}^e(\mathcal{L})$ and for a connected component $\Gamma_A$ of $\Gamma$ (with no univalent vertices or) whose univalent vertices are consecutive on one component $\mathcal{L}_A$ of $\mathcal{L}$, let $\Gamma^{\text{rev}}(A)$ denote the graph obtained from $\Gamma$ by reversing the order on the univalent vertices of $\Gamma_A$ induced by $i_\Gamma$.<fnref n="3" /> Set

$$I'(\Gamma, A) = \frac{1}{2} \int_{[0,1] \times F(A,L,\Gamma)} \bigwedge_{e \in E(\Gamma)} p_e^*(\tilde{\omega}(j_E(e))) ([\Gamma] - (-1)^{|E(\Gamma_A)|} [\Gamma^{\text{rev}}(A)]).$$

<footnote n="3">³When $\mathcal{L}_A$ is oriented, this order is a linear order if $U(\Gamma)$ has vertices of $V(\Gamma) \setminus A$ on $\mathcal{L}_A$; it is cyclic, otherwise. When $\mathcal{L}_A$ is not oriented, the order is not defined, but reversing the order is well-defined in any case. When $\mathcal{L}_A$ is oriented, we have $[\Gamma] = [\Gamma_A \# \mathcal{L}_A \Gamma_{V(\Gamma) \setminus A}]$ and $[\Gamma^{\text{rev}}(A)] = [\Gamma_A^{\text{rev}}(A) \# \mathcal{L}_A \Gamma_{V \setminus A}]$.</footnote>
Then we have

$$Z_n(1) - Z_n(0) = \sum_{(\Gamma, A) \in \mathcal{D}_{n}^{e,F}(\mathcal{L})} \zeta_{\Gamma} I'(\Gamma, A)$$

and

$$Z_{n,3n-2}(1) - Z_{n,3n-2}(0) = \sum_{(\Gamma,A) \in \mathcal{D}_{n,3n-2}^{e,F}(\mathcal{L})} \zeta_{\Gamma} I'(\Gamma, A).$$

PROOF OF THEOREM 7.30 ASSUMING PROPOSITION 9.7: There is a map rev from $\mathcal{A}(S^1)$ to itself that sends the class of a diagram $\Gamma$ to the class of the diagram obtained from $\Gamma$ by reversing the order of the univalent vertices on $S^1$ and by multiplying the class by $(-1)^{|T(\Gamma)|}$. The composition $w_C \circ \text{rev}$ equals $w_C$. Furthermore, $w_C$ sends odd-degree diagrams to zero, and $w_C$ is multiplicative with respect to the multiplication of $\mathcal{A}(S^1)$. So for any $(\Gamma, A) \in \mathcal{D}_{n,m}^{e,F}(S^1)$ as in the above statement such that $\Gamma$ has no component without univalent vertices, $(w_C([\Gamma]) - (-1)^{|E(\Gamma_A)|}w_C([\Gamma^{\text{rev}}(A)])$ is equal to

$$w_C([\Gamma_{V(\Gamma)\setminus A}]) (w_C([\Gamma_A]) - (-1)^{|E(\Gamma_A)|+|T(\Gamma_A)|} w_C(\text{rev}([\Gamma_A]))),$$

which is zero since $|E(\Gamma_A)| + |T(\Gamma_A)|$ is even when the degree of $\Gamma_A$ is even. So $w_C(I'(\Gamma, A))$ is always zero, and $w_C(\tilde{Z}_{n,3n-2}(\tilde{R}, K, (\omega(i))_{i \in \overline{3n-2}}))$ is independent of the chosen $\omega(i)$. The variation of $p^c(\tilde{Z}_{n,3n-2}(\tilde{R}, K, (\omega(i))_{i \in \overline{3n-2}}))$ is

$$\sum_{(\Gamma,A) \in \mathcal{D}_{n,3n-2}^{e,F}(S^1)} \zeta_{\Gamma} p^c \tilde{p}(I'(\Gamma, A)),$$

where $w_C$ sends $p^c \tilde{p}(I'(\Gamma, A))$ to zero, for the same reasons as above. $\square$

## 9.2 Sketch of proof of Proposition 9.2

According to Stokes' theorem, for any $\Gamma \in \mathcal{D}_{n,m}^e(\mathcal{L})$, where $m = 3n$ or $m = 3n-2$, we have

$$I(R, L, \Gamma, (\tilde{\omega}(i, 1))_{i \in \underline{m}}) = I(R, L, \Gamma, (\tilde{\omega}(i, 0))_{i \in \underline{m}}) + \sum_F \int_{[0,1] \times F} \bigwedge_{e \in E(\Gamma)} p_e^*(\tilde{\omega}(j_E(e))),$$

where the sum runs over the codimension-one faces $F$ of $C(R, L; \Gamma)$, which are described in Proposition 8.19. Let $\tilde{p}_e: C(R, L; \Gamma) \to C_e(R)$ be the natural restriction and set

$$p(\Gamma) = \mathbf{1}_{[0,1]} \times \prod_{e \in E(\Gamma)} \tilde{p}_e: [0,1] \times C(R, L; \Gamma) \to [0,1] \times \prod_{e \in E(\Gamma)} C_e(R).$$
For an edge $e_0$ of $E(\Gamma)$, let $p_{j_E(e_0)}: [0, 1] \times \prod_{e \in E(\Gamma)} C_e(R) \to [0, 1] \times C_2(R)$ be the composition of the natural projection onto $[0, 1] \times C_{e_0}(R)$ and the natural identification of $[0, 1] \times C_{e_0}(R)$ with $[0, 1] \times C_2(R)$. Define the form

$$ \Omega_{E(\Gamma)} = \bigwedge_{e \in E(\Gamma)} p_{j_E(e)}^* (\tilde{\omega}(j_E(e))) $$

on $[0, 1] \times \prod_{e \in E(\Gamma)} C_e(R)$. Set

$$ I(\Gamma, A) = \int_{[0,1] \times F(A,L,\Gamma)} p(\Gamma)^*(\Omega_{E(\Gamma)}) [\Gamma] $$

for any subset $A$ of $V(\Gamma)$ of cardinality at least 2, where $F(A, L, \Gamma)$ is empty (and hence $I(\Gamma, A) = 0$) if $A \cap U(\Gamma)$ is not a set of consecutive vertices on one component of $\mathcal{L}$. Set

$$ I(\Gamma, A, \infty) = \int_{[0,1] \times F_{\infty}(A,L,\Gamma)} p(\Gamma)^*(\Omega_{E(\Gamma)}) [\Gamma] $$

for any subset $A$ of $V(\Gamma)$ of cardinality at least 1, where $F_{\infty}(A, L, \Gamma)$ is empty (and hence $I(\Gamma, A, \infty) = 0$) if $A \cap U(\Gamma)$ is not empty. We have

$$ Z_{n,m}(1) - Z_{n,m}(0) = \sum_{\Gamma \in \mathcal{D}_{n,m}^e(\mathcal{L})} \left( \sum_{A \in \mathcal{P}_{\ge 2}(V(\Gamma))} \zeta_{\Gamma} I(\Gamma, A) + \sum_{A \in \mathcal{P}_{\ge 1}(V(\Gamma))} \zeta_{\Gamma} I(\Gamma, A, \infty) \right), $$

where $\mathcal{P}_{\ge 1}(V(\Gamma))$ denotes the set of the (nonstrict) nonempty subsets of $V(\Gamma)$. In order to prove Proposition 9.2, it suffices to prove that the codimension-one faces $F$ of the $C(R, L; \Gamma)$ that do not appear in the statement of Proposition 9.2 do not contribute.

This is the consequence of Lemmas 9.8 to 9.14, together with the analysis before Lemma 9.13.

**Lemma 9.8.** For any $\Gamma \in \mathcal{D}_{n,m}^e(\mathcal{L})$, For any nonempty subset $A$ of $V(\Gamma)$, we have $I(\Gamma, A, \infty) = 0$.

**PROOF:** Recall $F_{\infty}(A, L, \Gamma) = B_{\infty}(A, L, \Gamma) \times \tilde{S}(T_{\infty}R, A)$ from Section 8.4. Let $E_C$ be the set of the edges of $\Gamma$ that contain an element of $V(\Gamma) \setminus A$ and an element of $A$. Let $p_2$ denote the projection of $F_{\infty}(A, L, \Gamma)$ onto $\tilde{S}(T_{\infty}R, A)$. For $e \in E_A \cup E_C$, the map $P_e: (S^2)^{E_A \cup E_C} \to S^2$ is the projection onto the factor indexed by $e$. We prove that there exists a smooth map

$$ g: \tilde{S}(T_{\infty}R, A) \to (S^2)^{E_A \cup E_C} $$
such that $\bigwedge_{e \in E_A \cup E_C} p_e^*(\tilde{\omega}(j_E(e)))$ is equal to

$$ (\mathbf{1}_{[0,1]} \times (g \circ p_2))^* \left( \bigwedge_{e \in E_A \cup E_C} (\mathbf{1}_{[0,1]} \times P_e)^* (\tilde{\omega}_{S^2}(j_E(e))) \right). $$

If $e \in E_A \cup E_C$, then $p_e(F_\infty(A, L, \Gamma)) \subset \partial C_2(R) \setminus U(\check{R})$, and we have

$$ (\mathbf{1}_{[0,1]} \times p_e)^* (\tilde{\omega}(j_E(e))) = (\mathbf{1}_{[0,1]} \times (p_\tau \circ p_e))^* (\tilde{\omega}_{S^2}(j_E(e))). $$

If $e \in E_C$, then $p_\tau \circ p_e$ depends only on the projection on $S(T_\infty R)$ of the vertex at $\infty$ (of A). If $e \in E_A$, then $p_\tau \circ p_e$ factors through $\check{S}(T_\infty R, e)$. So in both cases, the map $p_\tau \circ p_e$ factors through $\check{S}(T_\infty R, A)$. Thus it may be expressed as $((P_e \circ g) \circ p_2)$. Therefore, if the degree of the form $\bigwedge_{e \in E_A \cup E_C} p_e^*(\tilde{\omega}_{S^2}(j_E(e)))$ is bigger than the dimension $3|A|$ of $[0, 1] \times \check{S}(T_\infty R, A)$, this form vanishes on $[0, 1] \times F_\infty(A, L, \Gamma)$. The degree of the form is $(2|E_A| + 2|E_C|)$, and we have

$$ 3|A| = 2|E_A| + |E_C|. $$

So the integral vanishes unless $E_C$ is empty. In this case, all the $p_\tau \circ p_e$, for $e \in E_A$ factor through the conjugates under the inversion $(x \mapsto x/\|x\|^2)$ of the translations that make sense, and the form $\bigwedge_{e \in E_A} p_e^*(\tilde{\omega}_{S^2}(j_E(e)))$ factors through the product by $[0, 1]$ of the quotient of $\check{S}(T_\infty R, A)$ by these translation conjugates. So it vanishes, too. $\square$

If there exists a smooth map from $[0, 1] \times F(A, L, \Gamma)$ to a manifold of strictly smaller dimension that factorizes the restriction of

$$ p(\Gamma) = (\mathbf{1}_{[0,1]} \times \prod_{e \in E(\Gamma)} \tilde{p}_e) : [0,1] \times C(R, L; \Gamma) \to [0,1] \times \prod_{e \in E(\Gamma)} C_e(R) $$

to $[0, 1] \times F(A, L, \Gamma)$, then we have $I(\Gamma, A) = 0$. We use this principle to get rid of some faces.

**Lemma 9.9.** Let $\Gamma \in \mathcal{D}_{n,m}^e(\mathcal{L})$. For any subset $A$ of $V(\Gamma)$ such that the graph $\Gamma_A$ defined before Proposition 9.2 is not connected and $\Gamma_A$ is not a pair of univalent vertices, we have $I(\Gamma, A) = 0$.

**PROOF:** In the fiber $\check{S}_A(T_{m(A)}\check{R}, L, \Gamma)$ of $F(A, L, \Gamma)$ we may translate one connected component of $\Gamma_A$ whose set of vertices is $C$, independently, without changing the restriction of $p(\Gamma)$ to $F(A, L, \Gamma)$. The translation vector is in $T_{m(A)}L$ when $C$ contains univalent vertices. Unless $C$ and $A \setminus C$ are reduced
to a univalent vertex, the quotient of $\tilde{S}_A(T_{m(A)}\tilde{R}, L, \Gamma)$ by these translations has a smaller dimension than $\tilde{S}_A(T_{m(A)}\tilde{R}, L, \Gamma)$, and $p(\Gamma)$ factors through the corresponding quotient of $[0, 1] \times F(A, L, \Gamma)$. $\square$

**Lemma 9.10.** Let $\Gamma \in \mathcal{D}_{n,m}^e(\mathcal{L})$. Let $A$ be a subset of $V(\Gamma)$ such that $|A| \ge 3$. If some trivalent vertex of $A$ belongs to exactly one edge of $\Gamma_A$, then $I(\Gamma, A) = 0$.

**PROOF:** Let $b$ be the mentioned trivalent vertex. Let $e$ be its edge in $\Gamma_A$, and let $d \in A$ be the other element of $e$. The group $]0, \infty[\Gamma$ acts on the map $t$ from $A$ to $T_{c(b)}R$ by moving $t(b)$ on the half-line from $t(d)$ through $t(b)$, by multiplying $(t(b) - t(d))$ by a scalar. When $|A| \ge 3$, this action defined on an open dense subset of $\tilde{S}_A(T_{m(A)}\tilde{R}, L, \Gamma)$ is not trivial, and $p(\Gamma)$ factors through the corresponding quotient of an open dense subset of $[0, 1] \times F(A, L, \Gamma)$, which is of smaller dimension. $\square$

## 9.3 Cancellations of nondegenerate faces

From now on, we will study cancellations that are no longer individual, and orientations must be seriously considered. Recall that the codimension-one faces are oriented as parts of the boundary of $C(R, L; \Gamma)$, with the outward normal first convention, where $C(R, L; \Gamma)$ is oriented by an orientation of $L$ and an order on $V(\Gamma)$. The relations between an orientation of $V(\Gamma)$, which orients $C(R, L; \Gamma)$, a vertex-orientation of $\Gamma$, and an edge-orientation of the set $H(\Gamma)$ of half-edges of $\Gamma$ are explained in Lemma 7.1, Corollary 7.2, and Remark 7.5. Fortunately, we do not have to fix everything to compare similar orientations.

**Lemma 9.11.** Let $\Gamma \in \mathcal{D}_{n,m}^e(\mathcal{L})$. Let $A$ be a subset of $V(\Gamma)$ such that at least one element of $A$ belongs to exactly two edges of $\Gamma_A$. Let $\mathcal{E}(\Gamma, A)$ denote the set of graphs of $\mathcal{D}_{n,m}^e(\mathcal{L})$ that are isomorphic to $\Gamma$ by an isomorphism that is only allowed to change the labels and the orientations of the edges of $\Gamma_A$. Such an isomorphism preserves $A$, and we have

$$ \sum_{\tilde{\Gamma}: \tilde{\Gamma} \in \mathcal{E}(\Gamma, A)} \zeta_{\tilde{\Gamma}} I(\tilde{\Gamma}, A) = 0. $$

**PROOF:** Let us first check that the isomorphisms of the statement preserve $A$. The vertices of the elements of $\mathcal{D}_{n,m}^e(\mathcal{L})$ are not numbered. A vertex is characterized by the half-edges that contain it. Therefore, the isomorphisms of the statement preserve the vertices of $V(\Gamma) \setminus A$. So they preserve $A$.
setwise. These isomorphisms also preserve the vertices that have adjacent edges outside $E(\Gamma_A)$ pointwise. The isomorphisms described below actually induce the identity map on $V(\Gamma)$.

Among the vertices of A that belong to exactly two edges of $\Gamma_A$ and one edge $j_E^{-1}(k)$ of $\Gamma$ outside $\Gamma_A$, choose the vertices such that $k$ is minimal. If there is one such vertex, then call this vertex $v_m$. Otherwise, there are two choices, and $v_m$ is chosen to be the vertex that belongs to the first half-edge of $j_E^{-1}(k)$.

We first describe an orientation-reversing diffeomorphism of the complement of a codimension-three submanifold of $F(A, L, \Gamma)$. Let $v_j$ and $v_k$ denote the (possibly equal) two other vertices of the two edges of $\Gamma_A$ that contain $v_m$. Consider the transformation $S$ of the space $\bar{S}_A(T_{c(A)}R)$ of nonconstant maps $f$ from A to $T_{c(A)}R$ up to translations and dilations that maps $f$ to $S(f)$, where

$$S(f)(v_\ell) = f(v_\ell) \text{ if } v_\ell \neq v_m, \text{ and} \\ S(f)(v_m) = f(v_j) + f(v_k) - f(v_m).$$

This is an orientation-reversing involution of $\bar{S}_A(T_{c(A)}R)$. The set of elements of $\bar{S}_A(T_{c(A)}R)$ whose image under $S$ is not in $\bar{S}_A(T_{c(A)}R)$ is a codimension-three submanifold of $\bar{S}_A(T_{c(A)}R)$. The fibered product of $S$ by the identity of the base $B(A, L, \Gamma)$ is an orientation-reversing smooth involution outside a codimension-three submanifold $F_S$ of $F(A, L, \Gamma)$. It is still denoted by $S$, as its product by $\mathbf{1}_{[0,1]}$ is, too.

Now, let $\sigma(A; \Gamma)(\tilde{\Gamma})$ be obtained from $(\tilde{\Gamma} \in \mathcal{E}(\Gamma, A))$ by exchanging the labels of the two edges of $\Gamma_A$ that contain $v_m$ and by reversing their orientations if (and only if) they both start or end at $v_m$. Then, as Figure 9.1 shows, we have

$$p(\tilde{\Gamma}) \circ S = p(\sigma(A; \Gamma)(\tilde{\Gamma})).$$

<figure><img src="image_80.png" /><figcaption>Figure 9.1: The parallelogram transformation $S$</figcaption></figure>

Recall

$$I(\Gamma, A) = \int_{[0,1] \times F(A,L,\Gamma)} p(\Gamma)^*(\Omega_{E(\Gamma)}) [\Gamma],$$
with the map

$$p(\Gamma) = \left( \mathbf{1}_{[0,1]} \times \prod_{e \in E(\Gamma)} \tilde{p}_e \right) : [0,1] \times C(R, L; \Gamma) \to [0,1] \times \prod_{e \in E(\Gamma)} C_e(R)$$

and $\Omega_{E(\Gamma)} = \bigwedge_{e \in E(\Gamma)} p_{j_E(e)}^*(\tilde{\omega}(j_E(e)))$. We have

$$\begin{align*}
I(\tilde{\Gamma}, A) &= \int_{[0,1] \times (F(A,L,\tilde{\Gamma}) \setminus F_S)} p(\tilde{\Gamma})^*(\Omega_{E(\Gamma)})[\tilde{\Gamma}] \\
&= -\int_{[0,1] \times (F(A,L,\tilde{\Gamma}) \setminus F_S)} S^* (p(\tilde{\Gamma})^*(\Omega_{E(\Gamma)})) [\tilde{\Gamma}] \\
&= -\int_{[0,1] \times (F(A,L,\tilde{\Gamma}) \setminus F_S)} (p(\tilde{\Gamma}) \circ S)^*(\Omega_{E(\Gamma)})[\tilde{\Gamma}] \\
&= -\int_{[0,1] \times (F(A,L,\tilde{\Gamma}) \setminus F_S)} p(\sigma(A; \Gamma)(\tilde{\Gamma}))^*(\Omega_{E(\Gamma)})[\tilde{\Gamma}] \\
&= -I(\sigma(A; \Gamma)(\tilde{\Gamma}), A)
\end{align*}$$

since we have $[\tilde{\Gamma}] = [\sigma(A; \Gamma)(\tilde{\Gamma})]$. Now, $\sigma(A; \Gamma)$ defines an involution of $\mathcal{E}(\Gamma, A)$, and we get

$$\sum_{\tilde{\Gamma} \in \mathcal{E}(\Gamma, A)} I(\tilde{\Gamma}, A) = \sum_{\tilde{\Gamma} \in \mathcal{E}(\Gamma, A)} I(\sigma(A; \Gamma)(\tilde{\Gamma}), A) = - \sum_{\tilde{\Gamma} \in \mathcal{E}(\Gamma, A)} I(\tilde{\Gamma}, A) = 0.$$

□

Maxim Kontsevich observed the symmetry of the above proof in [Kon94].

**Lemma 9.12.** Let $\Gamma \in \mathcal{D}_n^e(\mathcal{L})$. Let $A$ be a subset of $V(\Gamma)$ such that $\Gamma_A$ is a connected component of $\Gamma$. Let $\Gamma^{\text{eo}}(A)$ denote the graph obtained from $\Gamma$ by reversing all the orientations of the edges of $\Gamma_A$. Recall the notation of Proposition 9.2. If $\Gamma_A$ is a diagram of even degree without univalent vertices, then we have

$$I(\Gamma, A) + I(\Gamma^{\text{eo}}(A), A) = 0.$$

If $(\Gamma, A)$ is an element of the set $\mathcal{D}_{n,m}^{e,F}(\mathcal{L})$ defined in Proposition 9.2 and if $\Gamma_A$ has univalent vertices, let $\Gamma^{\text{eo,rev}}(A)$ denote the graph obtained from $\Gamma^{\text{eo}}(A)$ by reversing the order of the univalent vertices of $\Gamma_A$ on $\mathcal{L}_A$. Recall

$$I'(\Gamma, A) = \frac{1}{2} \int_{[0,1] \times F(A,L,\Gamma)} \bigwedge_{e \in E(\Gamma)} p_e^* (\tilde{\omega}(j_E(e))) ([\Gamma] - (-1)^{|E(\Gamma_A)|} [\Gamma^{\text{rev}}(A)])$$

from Proposition 9.7. Then we have

$$I(\Gamma, A) + I(\Gamma^{\text{eo,rev}}(A), A) = I'(\Gamma, A) + I'(\Gamma^{\text{eo,rev}}(A), A).$$
PROOF: Set $\bar{\Gamma}(A) = \Gamma^{\text{eo,rev}}(A)$ in both cases. The opposite of the identity map of $T_{c(A)}\tilde{R}$ induces a diffeomorphism from the fiber of $F(A, L, \Gamma)$ to the fiber of $F(A, L, \bar{\Gamma}(A))$, which induces a diffeomorphism from $F(A, L, \Gamma)$ to $F(A, L, \bar{\Gamma}(A))$ over the identity map of the base. Denote by $S$ the product of this diffeomorphism by $\mathbf{1}_{[0,1]}$. Let us carefully discuss orientations to determine when this diffeomorphism preserves the orientation.

Order and orient the vertices of $\Gamma$ so that the corresponding orientation of $H(\Gamma)$, as in Remark 7.5, is induced by the edge-orientation of $\Gamma$. There is a natural bijection from $V(\Gamma)$ to $V(\bar{\Gamma}(A))$. This bijection is the identity on the set $V(\Gamma) \setminus A$ of vertices of $\Gamma \setminus \Gamma_A$, unaffected by the modifications. When $A$ is not a pair of vertices in a $\theta$-component, a vertex of $\Gamma_A$ is characterized by the labels of the edges that contain it. Such a vertex is sent to the vertex of $\bar{\Gamma}(A)$ with the same set of labels of adjacent edges. When $A$ is a pair of vertices in a $\theta$-component, the vertex at which an edge of $\Gamma_A$ labeled by $i$ begins is sent to the vertex of $\bar{\Gamma}(A)$ at which the edge of $\bar{\Gamma}(A)$ labeled by $i$ ends.

Order the vertices of $\bar{\Gamma}(A)$ like the vertices of $\Gamma$ if $|E(\Gamma_A)|$ is even, and permute two vertices if $|E(\Gamma_A)|$ is odd. Orient $C(R, L; \Gamma)$ and $C(R, L; \bar{\Gamma}(A))$ with respect to the above orders of $V(\Gamma)$ and $V(\bar{\Gamma}(A))$, using the orientations of $R$ and $L$. Then $S$ reverses the orientation if and only if $|E(\Gamma_A)|$ is even since $|V(\Gamma_A)|$ is even.

Orient the vertices of $\bar{\Gamma}(A)$ like the vertices of $\Gamma$. So the orientation of $H(\bar{\Gamma}(A))$ associated to that vertex-orientation and to the above order of vertices is induced by the edge-orientation of $\bar{\Gamma}(A)$. When orientations are fixed as above, set $I_0(\Gamma, A) = \int_{[0,1]\times F(A,L,\Gamma)} p(\Gamma)^*(\Omega_{E(\Gamma)})$ so that $I(\Gamma, A) = I_0(\Gamma, A)[\Gamma]$. Let $e$ be an edge of $\Gamma$. Up to its edge-orientation, the edge $e$ is also an edge of $\bar{\Gamma}(A)$, and $p_{j_E(e)} \circ p(\bar{\Gamma}(A))$ restricts to $[0,1] \times F(A, L, \bar{\Gamma}(A))$ as $p_{j_E(e)} \circ p(\Gamma) \circ S^{-1}$. We have

$$ 
\begin{align*}
I_0(\bar{\Gamma}(A), A) &= \int_{[0,1] \times F(A,L,\bar{\Gamma}(A))} p(\bar{\Gamma}(A))^{*}(\Omega_{E(\bar{\Gamma}(A))}) \\
&= \int_{[0,1] \times F(A,L,\bar{\Gamma}(A))} \bigwedge_{e \in E(\bar{\Gamma})} (p_{j_E(e)} \circ p(\bar{\Gamma}(A)))^{*}(\tilde{\omega}(j_E(e))) \\
&= \int_{[0,1] \times F(A,L,\bar{\Gamma}(A))} (S^{-1})^{*} \left( \bigwedge_{e \in E(\bar{\Gamma})} (p_{j_E(e)} \circ p(\bar{\Gamma}))^{*} (\tilde{\omega}(j_E(e))) \right) \\
&= (-1)^{|E(\bar{\Gamma}_A)|+1} \int_{[0,1] \times F(A,L,\Gamma)} \bigwedge_{e \in E(\bar{\Gamma})} (p_{j_E(e)} \circ p(\bar{\Gamma}))^{*} (\tilde{\omega}(j_E(e))) \\
&= (-1)^{|E(\bar{\Gamma}_A)|+1} I_0(\Gamma, A).
\end{align*}
$$

In particular, if $\Gamma_A$ is a connected diagram of even degree without univalent vertices, then $|E(\Gamma_A)|$ is even, and $[\Gamma]$ is equal to $[\bar{\Gamma}(A)]$ and $[\Gamma^{\text{eo}}(A)]$. So we
have $I(\Gamma, A) = -I(\Gamma^{\text{eo}}(A), A)$. Otherwise, we have

$$ 
\begin{aligned}
I(\Gamma, A) + I(\bar{\Gamma}(A), A) &= \frac{1}{2} (I_0(\Gamma, A) - (-1)^{|E(\Gamma_A)|} I_0(\bar{\Gamma}(A), A)) [\Gamma] \\
&\quad + \frac{1}{2} (I_0(\bar{\Gamma}(A), A) - (-1)^{|E(\Gamma_A)|} I_0(\Gamma, A)) [\Gamma^{\text{rev}}(A)] \\
&= I'(\Gamma, A) + I'(\bar{\Gamma}(A), A).
\end{aligned}
$$

□

Lemmas 9.9, 9.10, and 9.11 allow us to get rid of the pairs $(\Gamma, A)$ with $|A| \ge 3$ such that

* at least one element of $A$ does not have all its adjacent edges in $E(\Gamma_A)$, or

* $\Gamma_A$ is disconnected.

Lemma 9.12 rules out the pairs $(\Gamma, A)$ such that $\Gamma_A$ is an even degree connected component of $\Gamma$, without univalent vertices (where $|A| \equiv 0 \mod 4$). Therefore, according to Lemma 9.8, we are left with

* the pairs $(\Gamma, A)$ of the statement of Proposition 9.2, for which $\Gamma_A$ is a connected component of $\Gamma$ (which may be an edge between two univalent vertices) and

* the following pairs, for which $|A| = 2$ and

- $\Gamma_A$ is an edge between two trivalent vertices,

- $\Gamma_A$ is an edge between a trivalent vertex and a univalent one, or

- $\Gamma_A$ is a pair of isolated consecutive univalent vertices,

since Lemma 9.9 rules out the disconnected $\Gamma_A$ with a trivalent vertex, and Lemma 9.11 rules out $\Gamma_A = \circled$.

The following lemma allows us to eliminate the pairs $(\Gamma, A)$ such that $\Gamma_A$ is an edge between two trivalent vertices using the Jacobi relation.

**Lemma 9.13.** The contributions to $(Z_{n,m}(1)-Z_{n,m}(0))$ of the faces $F(A, L, \Gamma)$ for which $\Gamma_A$ is an edge between two trivalent vertices cancel. More precisely, let $\Gamma \in \mathcal{D}_{n,m}^e(\mathcal{L})$. Let $A$ be a subset of $V(\Gamma)$ such that $\Gamma_A$ is an edge $e(\ell)$ with label $\ell$. Let $\Gamma/\Gamma_A$ be the labeled edge-oriented graph obtained from $\Gamma$ by contracting $\Gamma_A$ to one point, as in Figure 9.2. (The labels of the edges of $\Gamma/\Gamma_A$ belong to $\underline{m} \setminus \{\ell\}$. The graph $\Gamma/\Gamma_A$ has one four-valent vertex, and its other vertices are univalent or trivalent.) Let $\mathcal{E}(\Gamma; A)$ be the subset of $\mathcal{D}_{n,m}^e(\mathcal{L})$ that
contains the graphs $\tilde{\Gamma}$ equipped with a pair $A$ of vertices joined by an edge $e(\ell)$ with label $\ell$ such that $\tilde{\Gamma}/\tilde{\Gamma}_A$ is equal to $\Gamma/\Gamma_A$. Then we have

$$ \sum_{\tilde{\Gamma}: \tilde{\Gamma} \in \mathcal{E}(\Gamma; A)} \zeta_{\tilde{\Gamma}} I(\tilde{\Gamma}, A) = 0. $$

<figure><img src="image_81.png" /><figcaption>Figure 9.2: The graph $\Gamma$, its bold subgraph $\Gamma_A$, and $\Gamma/\Gamma_A$</figcaption></figure>

**PROOF:** Let us prove that there are 6 graphs in $\mathcal{E}(\Gamma; A)$. Let $a, b, c, d$ be the four half-edges of $\Gamma/\Gamma_A$ that contain its four-valent vertex. In $\tilde{\Gamma}$, the edge $e(\ell)$ goes from a vertex $v(\ell, 1)$ to a vertex $v(\ell, 2)$. The vertex $v(\ell, 1)$ is adjacent to the first half-edge of $e(\ell)$ and to two half-edges of $\{a, b, c, d\}$. The unordered pair of $\{a, b, c, d\}$ adjacent to $v(\ell, 1)$ determines $\tilde{\Gamma}$ as an element of $\mathcal{D}_{n,m}^e(\mathcal{L})$ and there are 6 graphs in $\mathcal{E}(\Gamma; A)$ labeled by the pairs of elements of $\{a, b, c, d\}$. They are $\Gamma = \Gamma_{ab}, \Gamma_{ac}, \Gamma_{ad}, \Gamma_{bc}, \Gamma_{bd}$, and $\Gamma_{cd}$.

The face $F(A, L, \Gamma)$ is fibered over $B(A, L, \Gamma)$ with fiber $\tilde{S}_A(T_{c(v(\ell,1))}R) =_{\tau} S^2$, which contains the direction of the vector from $c(v(\ell, 1))$ to $c(v(\ell, 2))$. Consistently order the vertices of the $\Gamma_.$ starting with $v(\ell, 1)$, $v(\ell, 2)$ (the other vertices are in natural correspondences for different $\Gamma_.$). Use these orders to orient the configuration spaces $C(R, L; \Gamma_.)$.

The oriented face $F(A, L, \Gamma_.)$ and the map

$$ p(\Gamma_{..}): [0,1] \times (F(A, L, \Gamma_{..}) \subset C(R, L; \Gamma_{..})) \to [0,1] \times \prod_{e \in E(\Gamma_{..})} C_2(R)^e $$

are the same for all the elements $\Gamma_.$ of $\mathcal{E}(\Gamma; A)$. Therefore, the

$$ I_0(\Gamma_{..}, A) = \int_{[0,1] \times F(A, L, \Gamma_{..})} p(\Gamma_{..})^*(\Omega_{E(\Gamma_{..})}) $$

are the same for all the elements $\Gamma_.$ of $\mathcal{E}(\Gamma; A)$ (for our consistent orders of the vertices), and the sum of the statement is

$$ \sum_{\tilde{\Gamma}: \tilde{\Gamma} \in \mathcal{E}(\Gamma; A)} \zeta_{\tilde{\Gamma}} I_0(\tilde{\Gamma}, A)[\tilde{\Gamma}] = \zeta_{\Gamma} I_0(\Gamma, A) \sum_{\tilde{\Gamma}: \tilde{\Gamma} \in \mathcal{E}(\Gamma; A)} [\tilde{\Gamma}]. $$

Let $e_1$ be the first half-edge of $e(\ell)$, and let $e_2$ be the other half-edge of $e(\ell)$. Equip $\Gamma = \Gamma_{ab}$ with a vertex-orientation, represented by $(a, b, e_1)$ at $v(\ell, 1)$,
and $(c, d, e_2)$ at $v(l, 2)$, which is consistent with its given edge-orientation (i.e., such that the edge-orientation of $H(\Gamma)$ is equivalent to its vertex-orientation, with respect to the above order of vertices). The orientation of $H(\Gamma)$ is represented by $(a, b, e_1, c, d, e_2, \dots)$. It induces the edge-orientation of $H(\Gamma)$, which is the same for all the elements of $\mathcal{E}(\Gamma; A)$.

Thus, permuting the letters $b, c, d$ cyclically gives rise to two other graphs $(\Gamma_{ac}$ and $\Gamma_{ad})$ in $\mathcal{E}(\Gamma; A)$ equipped with suitable vertex-orientations, respectively represented by

$$ (a, c, e_1) \text{ at } v(l, 1), \text{ and } (d, b, e_2) \text{ at } v(l, 2), \text{ or} \\ (a, d, e_1) \text{ at } v(l, 1), \text{ and } (b, c, e_2) \text{ at } v(l, 2). $$

The three other elements of $\mathcal{E}(\Gamma; A)$ with their suitable vertex-orientation are obtained from the three previous ones by exchanging the ordered pair before $e_1$ with the ordered pair before $e_2$. This does not change the unlabeled vertex-oriented graph. The first three graphs can be represented by the following three graphs identical outside the pictured disk:

<figure><img src="image_82.png" /></figure>

Then the sum $\sum_{\tilde{\Gamma}: \tilde{\Gamma} \in \mathcal{E}(\Gamma; A)} [\tilde{\Gamma}]$ is zero thanks to the Jacobi relation. $\square$

Now, we get rid of the remaining faces with the help of the STU relation.

**Lemma 9.14.** The contributions to $(Z_n(1) - Z_n(0))$ or to $(Z_{n,3n-2}(1) - Z_{n,3n-2}(0))$ of the faces $F(A, L, \Gamma)$ such that

* $\Gamma_A$ is an edge between a trivalent vertex and a univalent vertex or

* $A$ is a pair of consecutive univalent vertices and $\Gamma_A$ is not an edge of $\Gamma$

cancel. More precisely, let $\Gamma \in \mathcal{D}_{n,m}^e(\mathcal{L})$, let $A$ be a pair of consecutive univalent vertices of $\Gamma$ on a component of $\mathcal{L}$, and assume that $\Gamma_A$ is not an edge of $\Gamma$. Let $\Gamma/\Gamma_A$ be the labeled edge-oriented graph obtained from $\Gamma$ by contracting $\Gamma_A$ to one point. (The labels of the edges of $\Gamma/\Gamma_A$ belong to $\overline{m}$, $\Gamma/\Gamma_A$ has one bivalent vertex injected on $\mathcal{L}$.) Let $\mathcal{E}(\Gamma/\Gamma_A)$ be the subset of $\mathcal{D}_{n,m}^e(\mathcal{L})$ that contains the graphs $\tilde{\Gamma}$ equipped with a pair $A$ of vertices that are either

* two consecutive univalent vertices or

* a univalent vertex and a trivalent vertex connected by an edge,
such that $\Gamma/\Gamma_A$ is equal to $\tilde{\Gamma}/\tilde{\Gamma}_A$. If $m = 3n$ or if $m = 3n - 2$, then

$$ \sum_{\tilde{\Gamma}: \tilde{\Gamma} \in \mathcal{E}(\Gamma; A)} \zeta_{\tilde{\Gamma}} I(\tilde{\Gamma}, A) = 0. $$

**PROOF:** Note that the face $F(A, L, \Gamma)$ has two connected components if the only univalent vertices of $\Gamma$ on the component of $\mathcal{L}$ of the univalent vertices of $A$ are the two vertices of $A$. The two connected components correspond to the two possible linear orders of $A$ at the collapse.

Below, we consider these connected components as two different faces, and a face corresponds to a subset $A$ equipped with a linear order compatible with $i_{\Gamma}$. In particular, the graph and its face are determined by the labeled edge-oriented graph $\Gamma/\Gamma_A$ obtained from $\Gamma$ by contracting $A$ to one point, together with a linear order of the two half-edges of the bivalent vertex. Let $k \in \underline{m} \setminus j_E(E(\Gamma))$. Define $\Gamma_k^+$ (resp. $\Gamma_k^-$) to be the graph in $\mathcal{D}_{n,m}^e(\mathcal{L})$ with an edge $e(k)$ such that $j_E(e(k)) = k$, which goes from a univalent vertex $u$ to a trivalent vertex $t$ (resp. from a trivalent vertex $t$ to a univalent vertex $u$) forming a pair $A = \{u, t\}$ such that $\Gamma_k^+/(\Gamma_k^+)_A$ (resp. $\Gamma_k^-/(\Gamma_k^-)_A$) coincides with $\Gamma/\Gamma_A$.

Order the sets of vertices of the $\Gamma_k^{\pm}$ by putting the vertices of $A$ first, with respect to the order induced by the edge orientation (source first), and so that the order of the remaining vertices is the same for all $\Gamma_k^{\pm}$. Let $o(V(\Gamma)\setminus A)$ denote this order of $V(\Gamma)\setminus A$. For $(\tilde{\Gamma}, A) \in \mathcal{E}(\Gamma/\Gamma_A)$ such that $A$ is an ordered pair of univalent vertices of $\tilde{\Gamma}$, order $V(\tilde{\Gamma})$ by putting the vertices of $A$ first with respect to the linear order induced by the collapse, and next the others with the same order $o(V(\Gamma) \setminus A)$ as for the $\Gamma_k^{\pm}$.

Let $\phi: \mathbb{R}^3 \to \tilde{R}$ be an orientation-preserving diffeomorphism onto a neighborhood of the image of $A$ in a configuration of $F(A, L, \Gamma)$. There exists $\rho_{\phi}: \mathbb{R}^3 \to GL^+(\mathbb{R}^3)$ such that $T_x\phi(\rho_{\phi}(x)(\vec{v})) = \tau(\phi(x), \vec{v})$, for any $\vec{v} \in (\mathbb{R}^3 = T_x\mathbb{R}^3)$. Then for $\Gamma_k^+$, the configuration space is locally diffeomorphic to $\mathcal{L} \times \tilde{R} \times \dots$, where $\mathcal{L}$ contains the position $c(u) = \phi(x)$ of the univalent vertex $u$ of $A$, and $\tilde{R}$ contains the position $c(t)$ of the trivalent vertex $t$ of $A$. This position $c(t) = \phi(x + \lambda\rho_{\phi}(x)(\vec{v}))$ is described by a small positive $\lambda$, which plays the role of an inward normal near the collapse, and a vector $\vec{v}$ of $S^2$, which is equal to $p_{\tau} \circ p_{e(k)}(c)$ when $\lambda$ reaches 0. The face is diffeomorphic to $S^2 \times \mathcal{L} \times \dots$, where the projection onto the factor $S^2$ is $p_{\tau} \circ p_{e(k)}$, and the dots contain the coordinates of the remaining vertices, which are the same for all the considered diagrams.

For $\Gamma_k^-$, the configuration space is locally diffeomorphic to $\tilde{R} \times \mathcal{L} \times \dots$, where $\mathcal{L}$ contains the position $c(u) = \phi(x)$ of the univalent vertex $u$ of $A$, and $\tilde{R}$ contains the position $c(t)$ of the trivalent vertex $t$ of $A$. This position
$c(t) = \phi(x - \lambda \rho_{\phi}(x)(\vec{v}))$ is described by a small positive $\lambda$, which still plays the role of an inward normal near the collapse, and a vector $\vec{v}$ of $S^2$, which is $p_{\tau} \circ p_{e(k)}(c)$ when $\lambda$ reaches 0. The face is again diffeomorphic to $S^2 \times \mathcal{L} \times \dots$, where the projection onto the factor $S^2$ is $p_{\tau} \circ p_{e(k)}$, and the dots contain the coordinates of the remaining vertices, which are the same as for the $\Gamma_k^+$.

For $(\tilde{\Gamma}, A) \in \mathcal{E}(\Gamma/\Gamma_A)$ such that $A$ is an ordered pair of univalent vertices of $\tilde{\Gamma}$, the configuration space is locally diffeomorphic to $\mathcal{L} \times \mathcal{L} \times \dots$, where the first $\mathcal{L}$ contains the position $c(u_1) = \phi(x)$ of the first univalent vertex $u_1$ of $A$, and the second $\mathcal{L}$ contains the position $c(u_2)$ of the vertex $u_2$ that follows $u_1$ along $\mathcal{L}$. The position $c(u_2) = \phi(x + \lambda \rho_{\phi}(x)(\vec{t}))$ is again described by a small positive $\lambda$, which plays the role of an inward normal near the collapse, where $p_{\tau}$ maps the oriented unit tangent vector to $L$ at $c(u_1)$ to $\vec{t} \in S^2$ when $\lambda$ reaches 0. The face is diffeomorphic to $\mathcal{L} \times \dots$ with the same notation as before. So the previous faces are the products by $S^2$ of this one, and $p_{\tau} \circ p_{e(k)}$ is the projection to the factor $S^2$. Since the other $p_e$ do not depend on this factor $S^2$, $\int_{[0,1] \times F(A,L,\Gamma_k^+)} p(\Gamma_k^+)^*(\Omega_{E(\Gamma_k^+)})$ is equal to

$$ \int_{(t,c) \in [0,1] \times F(A,L,\Gamma)} \left( \int_{\{t\} \times \tilde{S}_{e(k)}(T_{c(A)}\tilde{R})} \tilde{\omega}(k) \right) p(\Gamma)^*(\Omega_{E(\Gamma)}), $$

where $\{t\} \times \tilde{S}_{e(k)}(T_{c(A)}\tilde{R})$ is the factor $S^2$ above. Furthermore, we have

$$ \int_{\{t\} \times \tilde{S}_{e(k)}(T_{c(A)}\tilde{R})} \tilde{\omega}(k) = 1 $$

since the integral of the closed form $\tilde{\omega}(k)$ over any representative of the homology class of the fiber of the unit tangent bundle of $\tilde{R}$ in $[0, 1] \times \partial C_2(R)$ is 1.

This argument, which also works for $\Gamma_k^-$, implies that all the integrals $I_0(\tilde{\Gamma}, A) = \int_{[0,1] \times F(A,L,\tilde{\Gamma})} p(\tilde{\Gamma})^*(\Omega_{E(\tilde{\Gamma})})$ coincide for all the $(\tilde{\Gamma}, A) \in \mathcal{E}(\Gamma/\Gamma_A)$ equipped with orders of their vertices as above. So it suffices to prove

$$ \zeta_{\Gamma}([\Gamma] + [\Gamma']) + \sum_{k \in \underline{m} \setminus j_{E}(E(\Gamma))} \zeta_{\Gamma_k^+}([\Gamma_k^+] + [\Gamma_k^-]) = 0, $$

where $\Gamma'$ is the graph obtained from $\Gamma$ by permuting the order of the two univalent vertices on $\mathcal{L}$, and all the graphs $\tilde{\Gamma}$ are vertex-oriented so that the vertex-orientation of $H(\tilde{\Gamma})$ induced by the fixed order of the vertices coincides with the edge-orientation of $H(\tilde{\Gamma})$ (as in Remark 7.5), for $m = 3n - 2$ and for $m = 3n$.

Let $a$ and $b$ denote the half-edges of $\Gamma$ that contain the vertices of $A$. Without loss of generality, assume that the vertex of $b$ follows the vertex
of a on $\mathcal{L}$ for $\Gamma$ (near the connected face). Let $o_V(H(\Gamma) \setminus \{a, b\})$ be an order of $H(\Gamma) \setminus \{a, b\}$, such that the order $(a, b, o_V(H(\Gamma) \setminus \{a, b\}))$ (i.e., $(a, b)$ followed by the elements $H(\Gamma) \setminus \{a, b\}$ ordered by $o_V(H(\Gamma) \setminus \{a, b\})$) induces the edge-orientation of $H(\Gamma)$.

Orient the vertices of $V(\Gamma) \setminus A$ in $\Gamma$ so that $o_V(H(\Gamma) \setminus \{a, b\})$ is induced by the order $o(V(\Gamma) \setminus A)$ and this vertex-orientation. Let $f$ (resp. $s$) denote the first (resp. second) half-edge of $e(k)$ in $\Gamma_k^{\pm}$. Then $(f, s, a, b, o_V(H(\Gamma) \setminus \{a, b\}))$ induces the edge-orientation of $H(\Gamma_k^{\pm})$. Equip the trivalent vertex of $A$ in $\Gamma_k^{\pm}$ with the vertex-orientation $((f \text{ or } s), a, b)$, which corresponds to the picture

$$ \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
  \node at (0,0) {$\begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
    \draw (0,0) -- (0.5,0);
    \draw (0,0) -- (-0.5,0);
    \draw (0,0) -- (0,0.5);
    \draw (0,0) -- (0,-0.5);
    \node at (0,0.25) {$\bullet$};
    \node at (0,-0.25) {$\bullet$};
  \end{tikzpicture}$};
  \node at (0.5,0) {$\bullet$};
  \node at (0.5,-0.5) {$\bullet$};
  \node at (0.5,0.25) {$a$};
  \node at (0.5,-0.25) {$a$};
  \node at (0.5,0.25) {$b$};
  \node at (0.5,-0.25) {$b$};
\end{tikzpicture}$, \qquad (9.1)

and equip the other vertices of $\Gamma_k^{\pm}$ with the same vertex-orientation as their vertex-orientation in $\Gamma$. Then the vertex-orientation of $H(\Gamma_k^+)$ is induced by $(f, s, a, b, o_V(H(\Gamma) \setminus \{a, b\}))$ and coincides with its edge-orientation. Similarly, the vertex-orientation of $H(\Gamma_k^-)$ is induced by $(f, a, b, s, o_V(H(\Gamma) \setminus \{a, b\}))$ and coincides with its edge-orientation. Thus, we have $[\Gamma_k^+] = [\Gamma_k^-]$ for any $k$, and $[\Gamma_k^+]$ is independent of $k$.

Note that $[\Gamma]$ looks like $\begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
  \draw (0,0) -- (0.5,0);
  \draw (0,0) -- (-0.5,0);
  \draw (0,0) -- (0,0.5);
  \draw (0,0) -- (0,-0.5);
  \node at (0,0.25) {$\bullet$};
  \node at (0,-0.25) {$\bullet$};
\end{tikzpicture}^b$ locally. It coincides with $[\Gamma_k^+]$ outside the pictured part. But $\Gamma'$ must be equipped with the opposite vertex-orientation, and $(-[\Gamma'])$ looks like $\begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
  \draw (0,0) -- (0.5,0);
  \draw (0,0) -- (-0.5,0);
  \draw (0,0) -- (0,0.5);
  \draw (0,0) -- (0,-0.5);
  \node at (0,0.25) {$\bullet$};
  \node at (0,-0.25) {$\bullet$};
\end{tikzpicture}^b$.

Thus, it suffices to prove

$$ \zeta_{\Gamma} \left( \left[ \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
  \draw (0,0) -- (0.5,0);
  \draw (0,0) -- (-0.5,0);
  \draw (0,0) -- (0,0.5);
  \draw (0,0) -- (0,-0.5);
  \node at (0,0.25) {$\bullet$};
  \node at (0,-0.25) {$\bullet$};
\end{tikzpicture} \right] - \left[ \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
  \draw (0,0) -- (0.5,0);
  \draw (0,0) -- (-0.5,0);
  \draw (0,0) -- (0,0.5);
  \draw (0,0) -- (0,-0.5);
  \node at (0,0.25) {$\bullet$};
  \node at (0,-0.25) {$\bullet$};
\end{tikzpicture} \right] \right) + 2(m - |E(\Gamma)|)\zeta_{\Gamma_k^+} \left[ \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
  \draw (0,0) -- (0.5,0);
  \draw (0,0) -- (-0.5,0);
  \draw (0,0) -- (0,0.5);
  \draw (0,0) -- (0,-0.5);
  \node at (0,0.25) {$\bullet$};
  \node at (0,-0.25) {$\bullet$};
\end{tikzpicture} \right] = 0 $$

if $m \in \{3n - 2, 3n\}$. With the expression of the $\zeta_{\Gamma}$ in Notation 7.16 and Notation 7.29, this equality is equivalent to the STU relation when $m > |E(\Gamma)|$. In particular, it is equivalent to the STU relation when $m = 3n$. When $m = 3n - 2$, if $m = |E(\Gamma)|$, then $\Gamma$ has exactly 2 univalent vertices, so we have $([\Gamma] + [\Gamma']) = 0$, and the equality is still true. $\square$

Proposition 9.2 is now proved. Proposition 9.7 follows from Proposition 9.2 and Lemma 9.12. $\square$

PROOF OF THEOREM 7.32: We follow the face cancellations in the proof of Proposition 9.2, to study the effect of changing $\omega_{S^2}(i) = \tilde{\omega}_{S^2}|_{\{0\} \times S^2}$ to $\omega'_{S^2}(i) = \tilde{\omega}_{S^2}|_{\{1\} \times S^2}$, for a closed two-form $\tilde{\omega}_{S^2}$ on $[0, 1] \times S^2$ as in Lemma 9.1, for some $i \in \mathbb{Z}$. We use the form $\omega(i) = p^*_{[0,1]\times S^2}(\tilde{\omega}_{S^2})$ on $[0, 1] \times C_2(S^3)$. Here, the involved graphs have no looped edges, 4 vertices, at most 3 edges, and hence at most one trivalent vertex. They are $\square$, $\ddagger$, and $\star$. The only cancellation that requires an additional argument is the cancellation of the
faces, for which $\Gamma$ is isomorphic to $\tilde{\mathcal{A}}$ and $A$ is a pair of univalent vertices of $\Gamma$. (The cancellation of Lemma 9.14 would involve $\mathcal{A}$.) In this case, the open face $F(A, L, \Gamma)$ is the configuration space of 2 vertices on the knot (one of them stands for the two vertices of $A$) and a trivalent vertex in $\mathbb{R}^3$. The integral $\int_{[0,1]\times F(A,L,\Gamma)} \bigwedge_{e \in E(\Gamma)} p_e^*(\omega(j_E(e)))$ is the pull-back of a 6-form on $[0,1] \times (S^2)^3$, by a map whose image is in the codimension 2 subspace of $[0,1] \times (S^2)^3$ in which two $S^2$-coordinates coincide. Indeed, the two edges that contain the vertices of $A$ have the same direction. Therefore, we have $I(\Gamma, A) = 0$ for these faces, and $w_C(\check{Z}_{2,3}(\mathbb{R}^3, K, (p_{S^2}^*(\omega_{S^2}(i)))_{i \in \check{\mathcal{I}}}))$ is independent of the chosen $\omega_{S^2}(i)$. Conclude with the arguments of Remark 7.31. $\square$
# Chapter 10

## First properties of Z and anomalies

### 10.1 Some properties of Z($\tilde{R}$, L, $\tau$)

Lemma 9.5 allows us to set

$$Z_n(\tilde{R}, L, \tau) = Z_n(\tilde{R}, L, (\omega(i)))$$

for any collection $(\omega(i))$ of homogeneous propagating forms of $(C_2(R), \tau)$, under the assumptions of Theorem 7.20. We still have to study how $Z_n(\tilde{R}, L, \tau)$ varies when $\tau$ varies inside its homotopy class when $L \neq \emptyset$, but the naturality of the construction of $Z_n$ already implies the following proposition.

**Proposition 10.1.** Let $R$ be the $Q$-sphere equipped with its neighborhood $\tilde{B}_{1,\infty}$ of $\infty$ of the beginning of Section 3.2. Let $\psi$ be an orientation-preserving diffeomorphism from $R$ to $\psi(R)$. Use the restriction of $\psi$ to the ball $\tilde{B}_{1,\infty}$ as an identification of $\tilde{B}_{1,\infty}$ with a neighborhood of $\psi(\infty)$ in $\psi(R)$. Define $\psi_*(\tau) = T\psi \circ \tau \circ (\psi^{-1} \times \mathbf{1}_{\mathbb{R}^3})$. Then we have

$$Z_n(\psi(\tilde{R}), \psi(L), \psi_*(\tau)) = Z_n(\tilde{R}, L, \tau)$$

for all $n \in \mathbb{N}$, where $p_1(\psi_*(\tau)) = p_1(\tau)$.

**PROOF:** The diffeomorphism $\psi$ induces natural diffeomorphisms $\psi_*$ from $C_2(R)$ to $C_2(\psi(R))$, and from the $\tilde{C}(R, L; \Gamma)$ to the $\tilde{C}(\psi(R), \psi(L); \Gamma)$. If $\omega$ is a homogeneous propagating form of $(C_2(R), \tau)$, then $(\psi_*^{-1})^*(\omega)$ is a homogeneous propagating form of $(C_2(\psi(R)), \psi_*(\tau))$ since the restriction of $(\psi_*^{-1})^*(\omega)$ to $U\psi(\tilde{R})$ is $(T\psi^{-1})^*(p_\tau^*(\omega_{S^2})) = (p_\tau \circ T\psi^{-1})^*(\omega_{S^2}) = p_{\psi_*(\tau)}^*(\omega_{S^2})$.
For any Jacobi diagram $\Gamma$ on the domain $\mathcal{L}$ of $L$, we have

$$ 
\begin{aligned}
I(\psi(R), \psi(L), \Gamma, (\psi_*^{-1})^*(\omega)) &= \int_{\check{C}(\psi(R), \psi(L); \Gamma)} \bigwedge_{e \in E(\Gamma)} p_e^*((\psi_*^{-1})^*(\omega)) \\
&= \int_{\check{C}(\psi(R), \psi(L); \Gamma)} (\psi_*^{-1})^* \left( \bigwedge_{e \in E(\Gamma)} p_e^*(\omega) \right) \\
&= I(R, L, \Gamma, \omega),
\end{aligned}
$$

where $\Gamma$ is equipped with an implicit orientation $o(\Gamma)$. Therefore, we have

$$ Z_n(\psi(\check{R}), \psi(L), \psi_*(\tau)) = Z_n(\check{R}, L, \tau) $$

for all $n \in \mathbb{N}$.

<figure><img src="image_83.png" /></figure>

We study some other properties of $Z_n(\check{R}, L, \tau)$.

Let $(\check{R}, \tau)$ be an asymptotic rational homology $\mathbb{R}^3$. Thanks to Lemma 9.6, $Z_n(\check{R}, \emptyset, \tau)$ depends only on the homotopy class of $\tau$ for any integer $n$. Set $Z_n(R, \tau) = Z_n(\check{R}, \emptyset, \tau)$ and $Z(R, \tau) = (Z_n(R, \tau))_{n \in \mathbb{N}}$. Using Notation 7.16, let $z_n(R, \tau) = p^c(Z_n(R, \tau))$ be the connected part of $Z_n(R, \tau)$, and set $z(R, \tau) = (z_n(R, \tau))_{n \in \mathbb{N}}$.

We give a direct elementary proof of the following proposition, which could also be proved in the same way as Corollary 10.4 below.

**Proposition 10.2.** For any propagating form $\omega$ of $C_2(R)$, we have

$$ Z(R, \omega) = \exp(z(R, \omega)). $$

In particular, for any asymptotically standard parallelization $\tau$ of $R$, we have

$$ Z(R, \tau) = \exp(z(R, \tau)). $$

**PROOF:** Let $\Gamma$ be a trivalent Jacobi diagram whose components are isomorphic to some $\Gamma_i$ for $i \in \underline{r}$ and such that $\Gamma$ has $k_i$ connected components isomorphic to $\Gamma_i$. Then we have

$$ I(R, \emptyset, \Gamma, (\omega)_{i \in \underline{3n}})[\Gamma] = \prod_{i=1}^{r} \left( I(R, \emptyset, \Gamma_i, (\omega)_{i \in \underline{3 \deg(\Gamma_i)}})^{k_i} [\Gamma_i]^{k_i} \right) $$

$$ \text{and } |\text{Aut}(\Gamma)| = \prod_{i=1}^{r} (k_i!(\text{Aut}(\Gamma_i))^{k_i}). \text{ Conclude with Proposition 7.25.} \quad \square $$

<figure><img src="image_84.png" /></figure>

Recall the coproduct maps $\Delta_n$ defined in Section 6.5.

**Proposition 10.3.** Under the assumptions of Theorem 7.20, we have

$$ \Delta_n(Z_n(\check{R}, L, \tau)) = \sum_{i=0}^{n} Z_i(\check{R}, L, \tau) \otimes Z_{n-i}(\check{R}, L, \tau). $$
PROOF: Let $T_i = Z_i(R, L, \tau) \otimes Z_{n-i}(R, L, \tau)$. We have

$$T_i = \sum \frac{1}{|\mathrm{Aut}(\Gamma')|} I(\Gamma', (\omega)_{j \in \underline{3i}}) \frac{1}{|\mathrm{Aut}(\Gamma'')|} I(\Gamma'', (\omega)_{j \in \underline{3(n-i)}}) [\Gamma'] \otimes [\Gamma''],$$

where the sum runs over the pairs $(\Gamma', \Gamma'')$ in $\mathcal{D}_i^u(\mathcal{L}(L)) \times \mathcal{D}_{n-i}^u(\mathcal{L}(L))$. Use Remark 7.26 to view the summands as a measure of configurations of graphs $\Gamma' \sqcup \Gamma''$ (which may correspond to several elements of $\mathcal{D}_n^u(\mathcal{L}(L))$) together with a choice of an embedded subgraph $\Gamma'$. $\square$

**Corollary 10.4.** Recall the projection $p^c$ of Corollary 6.40 from $\check{\mathcal{A}}(S^1)$ to the space $\check{\mathcal{A}}^c(S^1)$ of its primitive elements. If $L$ has one component, set

$$\check{z}(R, L, \tau) = p^c(\check{Z}(\check{R}, L, \tau)).$$

Then we have

$$\check{Z}(\check{R}, L, \tau) = \exp(\check{z}(R, L, \tau)).$$

PROOF: This is a direct consequence of Lemma 6.37, Proposition 10.3 and Theorem 6.41. $\square$

## 10.2 On the anomaly $\beta$

We now study how $z_n(R, \tau)$, which is defined before Proposition 10.2, depends on $\tau$.

**Definition 10.5.** Let $\rho: (B^3, \partial B^3) \to (SO(3), 1)$ be the map of Definition 4.4. Extend it to $\mathbb{R}^3$ by considering $B^3$ as the unit ball of $\mathbb{R}^3$ and by letting $\rho$ map $(\mathbb{R}^3 \setminus B^3)$ to 1. Consider the parallelization $\tau_s \circ \psi_{\mathbb{R}}(\rho)$, where $\psi_{\mathbb{R}}(\rho)(x, v) = (x, \rho(x)(v))$. Set

$$\beta_n = z_n(S^3, \tau_s \circ \psi_{\mathbb{R}}(\rho)).$$

**Proposition 10.6.** Let $(\check{R}, \tau_0)$ be an asymptotic rational homology $\mathbb{R}^3$, and let $\tau_1$ be a parallelization of $\check{R}$ that coincides with $\tau_0$ outside $B_R$. Then we have

$$z_n(R, \tau_1) - z_n(R, \tau_0) = \frac{p_1(\tau_1) - p_1(\tau_0)}{4} \beta_n$$

for any integer $n$.
Proposition 10.6 is an easy consequence of Proposition 10.7 below. The latter looks more complicated, but it is very useful since it offers more practical definitions of the anomaly

$$ \beta = (\beta_n)_{n \in \mathbb{N}} $$

when applied to $(\check{R}, \tau_0, \tau_1) = (\mathbb{R}^3, \tau_s, \tau_s \circ \psi_{\mathbb{R}}(\rho))$ (and to the case in which $\tilde{\omega}_{S^2}(i)$ is the pull-back of $\omega_{0,S^2}(i)$ under the natural projection from $[0, 1] \times S^2$ to $S^2$).

**Proposition 10.7.** Let $(\check{R}, \tau_0)$ be an asymptotic rational homology $\mathbb{R}^3$, and let $\tau_1$ be a parallelization of $\check{R}$ that coincides with $\tau_0$ outside $B_R$. For $i \in \underline{3n}$, let $\omega_{0,S^2}(i)$ and $\omega_{1,S^2}(i)$ be two volume-one forms on $S^2$. Then there exists a closed two-form $\tilde{\omega}_{S^2}(i)$ on $[0, 1] \times S^2$ such that the restriction of $\tilde{\omega}_{S^2}(i)$ to $\{t\} \times S^2$ is $\omega_{t,S^2}$ for $t \in \{0, 1\}$. For any such forms $\tilde{\omega}_{S^2}(i)$, there exist closed 2-forms $\tilde{\omega}(i)$ on $[0, 1] \times \check{R}$ such that

* the restriction of $\tilde{\omega}(i)$ to $\{t\} \times \check{R}$ is $p_{\tau_t}^*(\omega_{t,S^2}(i))$ for $t \in \{0, 1\}$,

* the restriction of $\tilde{\omega}(i)$ to $[0, 1] \times (U(\check{R} \setminus B_R))$ is $(\mathbf{1}_{[0,1]} \times p_{\tau_0})^*(\tilde{\omega}_{S^2}(i))$.

Then we have

$$ \begin{aligned} z_n(R, \tau_1) - z_n(R, \tau_0) &= z_n([0, 1] \times UB_R; (\tilde{\omega}(i))_{i \in \underline{3n}}) \\ &= \sum_{\Gamma \in \mathcal{D}_n^c} \zeta_{\Gamma} \int_{[0,1] \times \check{S}_{V(\Gamma)}(TB_R)} \bigwedge_{e \in E(\Gamma)} p_e^*(\tilde{\omega}(j_E(e))) [\Gamma] \\ &= \frac{p_1(\tau_1) - p_1(\tau_0)}{4} \beta_n, \end{aligned} $$

and $\beta_n = 0$ if $n$ is even. (Recall that the orientation of $\check{S}_{V(\Gamma)}(T\check{R})$ is defined in Lemma 9.3.)

**PROOF:** The existence of $\tilde{\omega}_{S^2}(i)$ comes from Lemma 9.1. In order to prove the existence of $\tilde{\omega}(i)$, which is defined on $\partial([0, 1] \times S^2 \times B_R)$ by the conditions, we need to extend it to $[0, 1] \times S^2 \times B_R$. The obstruction belongs to

$$ H^3([0, 1] \times S^2 \times B_R, \partial([0, 1] \times S^2 \times B_R)) \cong H_3([0, 1] \times S^2 \times B_R), $$

which is trivial. So $\tilde{\omega}(i)$ extends as desired. In order to prove that the first equality is a consequence of Corollary 9.4, extend the forms $\tilde{\omega}(i)$ of the statement to $[0, 1] \times C_2(R)$ as forms that satisfy the conditions in Proposition 9.2. First extend the $\tilde{\omega}(i)$ to $[0, 1] \times (\partial C_2(R) \setminus UB_R)$ as $(\mathbf{1}_{[0,1]} \times p_{\tau_0})^*(\tilde{\omega}_{S^2}(i))$. Next extend the restriction of $\tilde{\omega}(i)$ to $\{0\} \times \partial C_2(R)$ (resp. to $\{1\} \times \partial C_2(R)$) on
$\{0\} \times C_2(R)$ (resp. on $\{1\} \times C_2(R)$) as a propagating form of $(C_2(R), \tau_0)$ (resp. of $(C_2(R), \tau_1)$) as in Section 3.3. Thus, $\tilde{\omega}(i)$ is defined consistently on $\partial([0, 1] \times C_2(R))$, and it extends as a closed form that satisfies the assumptions in Proposition 9.2 as in Lemma 9.1. Corollary 9.4, Lemma 9.6, and Lemma 9.12 yield

$$z_n(R, \tau_1) - z_n(R, \tau_0) = \sum_{\Gamma \in \mathcal{D}_n^c} \frac{1}{(3n)!2^{3n}} \int_{[0,1] \times \tilde{S}_{V(\Gamma)}(TB_R)} \bigwedge_{e \in E(\Gamma)} p_e^*(\tilde{\omega}(j_E(e))) [\Gamma],$$

which is zero if $n$ is even. So everything is proved when $n$ is even. Assume that $n$ is odd.

There exists a map $g: (\tilde{R}, \tilde{R} \setminus B_R) \to (SO(3), 1)$ such that $\tau_1 = \tau_0 \circ \psi_R(g)$. Using $\tau_0$ to identify $\tilde{S}_{V(\Gamma)}(T\tilde{R})$ with $\tilde{R} \times \tilde{S}_{V(\Gamma)}(\mathbb{R}^3)$ makes clear that $(z_n(R, \tau_0 \circ \psi_R(g)) - z_n(R, \tau_0))$ does not depend on $\tau_0$. For any $g: (\tilde{R}, \tilde{R} \setminus B_R) \to (SO(3), 1)$, set $z'_n(g) = z_n(R, \tau_0 \circ \psi_R(g)) - z_n(R, \tau_0)$. Then $z'_n$ is a homomorphism from $[(B_R, \partial B_R), (SO(3), 1)]$ to the vector space $\mathcal{A}_n^c(\emptyset)$ over $\mathbb{R}$. Theorem 4.5 and Lemma 4.6 yield

$$z'_{n}(g) = \frac{\deg(g)}{2} z'_{n}(\rho_{B_{R}}(B^{3})).$$

It is easy to see that $z'_n(\rho_{B_R}(B^3))$ is independent of $\tilde{R}$. Since Example 7.18 shows $z_n(S^3, \tau_s) = 0$, we have $z'_n(\rho_{B_R}(B^3)) = \beta_n$ by Definition 10.5. Recall $p_1(\tau_0 \circ \psi_R(g)) - p_1(\tau_0) = 2 \deg(g)$ from Theorem 4.5. $\square$

**Remark 10.8.** The anomaly $\beta$ is the opposite of the constant $\xi$ defined in [Les04a, Section 1.6].

**Corollary 10.9.** Let $(\tilde{R}, \tau)$ be an asymptotic rational homology $\mathbb{R}^3$, then

$$z_n(R, \tau) - \frac{p_1(\tau)}{4} \beta_n$$

is independent of $\tau$. Set $\mathfrak{z}_n(R) = z_n(R, \tau) - \frac{p_1(\tau)}{4} \beta_n$, $\mathfrak{z}(R) = (\mathfrak{z}_n(R))_{n \in \mathbb{N}}$, and

$$\mathcal{Z}(R) = \exp(\mathfrak{z}(R)).$$

Then

$$\mathcal{Z}(R) = Z(R, \tau) \exp \left( - \frac{p_1(\tau)}{4} \beta \right)$$

is the invariant $\mathcal{Z}(R, \emptyset)$ that was announced in Theorem 7.20.

**PROOF:** See Proposition 10.2. $\square$
**Proposition 10.10.** The degree-one part of the anomaly $\beta$ is $\beta_1 = \frac{1}{12}[\Theta]$.

**PROOF:** According to Proposition 7.17, we have $z_1(R, \tau) = \frac{\Theta(R, \tau)}{12}[\Theta]$. Proposition 10.6 implies $z_1(R, \tau_1) - z_1(R, \tau_0) = \frac{p_1(\tau_1) - p_1(\tau_0)}{4}\beta_1$, while Corollary 4.8 implies

$$ \Theta(R, \tau_1) - \Theta(R, \tau_0) = \frac{1}{4}(p_1(\tau_1) - p_1(\tau_0)). $$

□

**Corollary 10.11.** Let $(\tilde{R}, \tau)$ be an asymptotic rational homology $\mathbb{R}^3$, then we have $Z_1(R, \tau) = z_1(R, \tau) = \frac{\Theta(R, \tau)}{12}[\Theta]$ and

$$ Z_1(R) = \mathfrak{z}_1(R) = \frac{\Theta(R)}{12}[\Theta] $$

in $\mathcal{A}_1(\emptyset) = \mathbb{R}[\Theta]$.

**PROOF:** The first equality is a direct consequence of Proposition 7.17. The second one follows from Corollary 4.8, Corollary 10.9, and Proposition 10.10.

□

**Remark 10.12.** According to Proposition 10.7, the even-degree part of the anomaly $\beta$ vanishes. The values of $\beta_{2n+1}$ are unknown when $n \ge 1$. We may hope them to be zero, but I do not know any conjecture about them.

## 10.3 On the anomaly $\alpha$

We define the *anomaly*

$$ \alpha = (\alpha_n)_{n \in \mathbb{N}}, $$

which is sometimes called the *Bott–Taubes anomaly*, below. Let $v \in S^2$. Let $D_v$ denote the linear map

$$ D_v: \begin{array}{ccc} \mathbb{R} & \to & \mathbb{R}^3 \\ 1 & \mapsto & v. \end{array} $$

Let $\mathcal{D}_n^c(\mathbb{R})$ denote the set of degree $n$, connected, $3n-2$-numbered diagrams on $\mathbb{R}$ with at least one univalent vertex, without looped edges. As in Definition 7.6, a degree $n$ diagram $\tilde{\Gamma}$ is *numbered* if the edges of $\tilde{\Gamma}$ are oriented and if $E(\tilde{\Gamma})$ is equipped with an injection $j_E: E(\tilde{\Gamma}) \hookrightarrow 3n-2$, which numbers its edges. Let $\tilde{\Gamma} \in \mathcal{D}_n^c(\mathbb{R})$. Define $\tilde{C}(D_v; \tilde{\Gamma})$ as in Section 7.1, where the
line $D_v$ of $\mathbb{R}^3$ replaces the link $L$ of $\check{R}$, and $\mathbb{R}$ replaces the domain $\mathcal{L}$. Let $\check{Q}(v; \check{\Gamma})$ be the quotient of $\check{C}(D_v; \check{\Gamma})$ by the translations parallel to $D_v$ and by the dilations. Then $\check{Q}(v; \check{\Gamma})$ is the space denoted by $\check{\mathcal{S}}(\mathbb{R}^3, v; \check{\Gamma})$ before Lemma 8.16. Let $Q(v; \check{\Gamma})$ denote the closure of $\check{Q}(v; \check{\Gamma})$ in $S_{V(\check{\Gamma})}(\mathbb{R}^3)$. According to Lemma 8.16, the space $Q(v; \check{\Gamma})$, which coincides with $\mathcal{S}(\mathbb{R}^3, v; \check{\Gamma})$, is a compact smooth manifold with ridges.

To each edge $e$ of $\check{\Gamma}$, associate a map $p_{e,S^2}$, which maps a configuration of $\check{Q}(v; \check{\Gamma})$ to the direction of the vector from the origin of $e$ to its end in $S^2$. This map extends to $Q(v; \check{\Gamma})$ according to Theorem 8.12.

Now, define $\check{Q}(\check{\Gamma})$ (resp. $Q(\check{\Gamma})$) as the total space of the fibration over $S^2$ whose fiber over $v$ is $\check{Q}(v; \check{\Gamma})$ (resp. $Q(v; \check{\Gamma})$). The configuration space $\check{Q}(\check{\Gamma})$ and its compactification $Q(\check{\Gamma})$ carry natural smooth structures. The configuration space $\check{Q}(\check{\Gamma})$ is oriented as follows, when a vertex-orientation $o(\check{\Gamma})$ is given. Equip $\check{C}(D_v; \check{\Gamma})$ with its orientation induced by Corollary 7.2, as before. Orient $\check{Q}(v; \check{\Gamma})$ so that $\check{C}(D_v; \check{\Gamma})$ is locally homeomorphic to the oriented product (translation vector $z$ in $\mathbb{R}v$, ratio of homothety $\lambda \in ]0, \infty[$) $\times \check{Q}(v; \check{\Gamma})$. Next orient $\check{Q}(\check{\Gamma})$ with the (base($= S^2$) $\oplus$ fiber) convention. (We can summarize this by saying that the $S^2$-coordinates replace $(z, \lambda)$.)

**Proposition 10.13.** For $i \in \underline{3n-2}$, let $\omega(i, S^2)$ be a volume-one form of $S^2$. Define

$$I(\check{\Gamma}, o(\check{\Gamma}), \omega(i, S^2)) = \int_{\check{Q}(\check{\Gamma})} \bigwedge_{e \in E(\check{\Gamma})} p_{e,S^2}^*(\omega(j_E(e), S^2)).$$

Set

$$\alpha_n = \frac{1}{2} \sum_{\check{\Gamma} \in \mathcal{D}_n^c(\mathbb{R})} \zeta_{\check{\Gamma}} I(\check{\Gamma}, o(\check{\Gamma}), \omega(i, S^2)) [\check{\Gamma}, o(\check{\Gamma})] \in \mathcal{A}(\mathbb{R}),$$

where $\zeta_{\check{\Gamma}} = \frac{(3n-2-|E(\check{\Gamma})|)!}{(3n-2)!2^{|E(\check{\Gamma})|}}$. Then $\alpha_n$ does not depend on the chosen $\omega(i, S^2)$, we have $\alpha_1 = \frac{1}{2} [\zeta], \text{ and } \alpha_{2n} = 0 \text{ for all } n$.

**PROOF:** Let us first prove that $\alpha_n$ does not depend on the chosen $\omega(i, S^2)$, by proving that its variation vanishes when $\omega(i, S^2)$ is changed to some $\tilde{\omega}(i, 1, S^2)$. According to Lemma 9.1, there exists a closed 2-form $\tilde{\omega}(i, S^2)$ on $[0, 1] \times S^2$ whose restriction to $\{0\} \times S^2$ is $\omega(i, S^2) = \tilde{\omega}(i, 0, S^2)$ and whose restriction to $\{1\} \times S^2$ is $\tilde{\omega}(i, 1, S^2)$. According to Stokes' theorem, for any $\check{\Gamma} \in \mathcal{D}_n^c(\mathbb{R})$, we have

$$I(\check{\Gamma}, o(\check{\Gamma}), (\tilde{\omega}(i, 1, S^2))_{i \in \underline{3n-2}}) - I(\check{\Gamma}, o(\check{\Gamma}), (\tilde{\omega}(i, 0, S^2))_{i \in \underline{3n-2}}) \\ = \sum_F \int_{[0,1] \times F} \bigwedge_{e \in E(\check{\Gamma})} p_{e,S^2}^*(\tilde{\omega}(j_E(e), S^2)),$$
where $p_{e,S^2}: [0,1] \times \check{Q}(\check{\Gamma}) \to [0,1] \times S^2$ denotes the product by $\mathbf{1}_{[0,1]}$ of $p_{e,S^2}$, and the sum runs over the codimension-one faces $F$ of $Q(\check{\Gamma})$. These faces fiber over $S^2$, and the fibers over $v \in S^2$ are the codimension-one faces $f(A, v; \check{\Gamma})$ of $Q(v, \check{\Gamma}) = S(\mathbb{R}^3, v; \check{\Gamma})$ for the strict subsets $A$ of $V(\check{\Gamma})$ with cardinality at least 2 whose univalent vertices are consecutive on $\mathbb{R}$, as in Lemma 8.17. Let $F(A, \check{\Gamma})$ denote the face with fiber $f(A, v; \check{\Gamma})$. Now, it suffices to prove that the contributions of all the $F(A, \check{\Gamma})$ vanish.

When the product of all the $p_{e,S^2}$ factors through a quotient of $[0,1] \times F(A, \check{\Gamma})$ of smaller dimension, the face $F(A, \check{\Gamma})$ does not contribute. This allows us to get rid of

* the faces $F(A, \check{\Gamma})$ for which $\check{\Gamma}_A$ is not connected and $A$ is not a pair of univalent vertices of $\check{\Gamma}$, as in Lemma 9.9, and

* the faces $F(A, \check{\Gamma})$ for which $|A| \ge 3$ and $\check{\Gamma}_A$ has a univalent vertex that was trivalent in $\check{\Gamma}$, as in Lemma 9.10.

We also have faces that cancel each other for graphs that are identical outside their $\check{\Gamma}_A$ part.

* The faces $F(A, \check{\Gamma})$ (which are not already listed) such that $\check{\Gamma}_A$ has at least one bivalent vertex cancel (by pairs) by the parallelogram identification as in Lemma 9.11.

* The faces $F(A, \check{\Gamma})$ for which $\check{\Gamma}_A$ is an edge between two trivalent vertices, cancel by triples, thanks to the Jacobi relation as in Lemma 9.13.

* Similarly, two faces for which $A$ is a pair of (necessarily consecutive in $\mathbb{R}$) univalent vertices of $\check{\Gamma}$, cancel $(3n - 2 - |E(\Gamma)|)$ faces $F(\check{\Gamma}', A')$ for which $\check{\Gamma}'_{A'}$ is an edge between a univalent vertex of $\check{\Gamma}$ and a trivalent vertex of $\check{\Gamma}$, thanks to the STU relation (and to Lemma 6.26) as in Lemma 9.14.

Here, there are no faces left, and $\alpha_n$ does not depend on the chosen $\omega(i, S^2)$. The computation of $\alpha_1$ is straightforward.

Let us prove that $\alpha_n = 0$ for any even $n$. Let $\check{\Gamma}$ be a numbered graph, and let $\check{\Gamma}^{eo}$ be obtained from $\check{\Gamma}$ by reversing the orientations of the $|E|$ edges of $\check{\Gamma}$. Consider the map $r$ from $\check{Q}(\check{\Gamma}^{eo})$ to $\check{Q}(\check{\Gamma})$ that composes a configuration by the multiplication by $(-1)$ in $\mathbb{R}^3$. It sends a configuration over $v \in S^2$ to a configuration over $(-v)$. It is a fibered space map over the orientation-reversing antipode of $S^2$. Equip $\check{\Gamma}$ and $\check{\Gamma}^{eo}$ with the same vertex-orientation and with the same orders on their vertex sets. Then our map $r$ is orientation-preserving if and only if $|T(\check{\Gamma})| + 1$ is even. The vertex-orientations of $H(\check{\Gamma})$
and $H(\check{\Gamma}^{eo})$ can be consistent with both of the edge-orientations of $H(\check{\Gamma})$ and $H(\check{\Gamma}^{eo})$ if and only if $|E(\check{\Gamma})|$ is even. We have $p_{e,S^2,\check{\Gamma}^{eo}} = p_{e,S^2,\check{\Gamma}} \circ r$ for all the edges $e$ of $\check{\Gamma}^{eo}$. Since $|E(\check{\Gamma})| = n + |T(\check{\Gamma})|$, we get

$$I(\check{\Gamma}^{eo}, \omega(i, S^2)) [\check{\Gamma}^{eo}] = (-1)^{n+1} I(\check{\Gamma}, \omega(i, S^2)) [\check{\Gamma}]. \quad \square$$

<figure><img src="image_85.png" /></figure>

**Note 10.14.** It is known that $\alpha_3 = 0$ [Poi02, Proposition 1.4]. Sylvain Poirier also found that $\alpha_5 = 0$ with the help of a Maple program. Furthermore, according to [Les02, Corollary 1.4], $\alpha_{2n+1}$ is a combination of diagrams with two univalent vertices, and $\mathcal{Z}(S^3, L) = \check{\mathcal{Z}}(S^3, L)$ is obtained from the Kontsevich integral $Z^K$ by inserting $d$ times the plain part of $2\alpha$ on each degree $d$ connected component of a diagram. A similar statement is valid for the functorial extension of $\mathcal{Z}$ to tangles described in the book's third part. See Corollary 12.29 and Lemma 13.19, which provides an alternative definition of $\alpha$. It has been conjectured that $\alpha_n = 0$ for $n > 1$ and incorrect proofs have circulated.

## 10.4 Dependence on the forms for straight links

In this section, we prove Theorems 7.40 and 7.42. In order to do it, we will prove the following lemma.

**Lemma 10.15.** Under the assumptions of Theorem 7.40, the element

$$Z_{n,3n-2}(\check{R}, L, (\omega(i))_{i \in \overline{3n-2}}) = \sum_{\Gamma \in \mathcal{D}_{n,3n-2}^e(\mathcal{L})} \zeta_{\Gamma} I(R, L, \Gamma, (\omega(i))_{i \in \overline{3n-2}}) [\Gamma]$$

of $\mathcal{A}_n(\mathcal{L})$ is independent of the chosen propagating forms $\omega(i)$ of $(C_2(R), \tau)$.

**Lemma 10.16.** Lemma 10.15 implies Theorem 7.40 and Theorem 7.42.

**PROOF:** Let us prove that Lemma 10.15 implies Theorem 7.40. $\mathcal{A}_n(\mathcal{L}) = \mathcal{A}_n(\emptyset) \oplus \mathcal{A}'_n(\mathcal{L})$, where $\mathcal{A}'_n(\mathcal{L})$ is the subspace of $\mathcal{A}_n(\mathcal{L})$ generated by the Jacobi diagrams with at least one univalent vertex. Since we know from Lemma 9.6 that $Z_n(\check{R}, \emptyset, (\omega(i))_{i \in \overline{3n}})$ is independent of the chosen propagating forms $\omega(i)$ of $(C_2(R), \tau)$, we focus on the projection $Z'_n(\check{R}, L, (\omega(i))_{i \in \overline{3n}})$ of

$$Z_n(\check{R}, L, (\omega(i))_{i \in \overline{3n}}) = \sum_{\Gamma \in \mathcal{D}_n^e(\mathcal{L})} \zeta_{\Gamma} I(R, L, \Gamma, (\omega(i))_{i \in \overline{3n}}) [\Gamma] \in \mathcal{A}_n(\mathcal{L})$$
onto $\mathcal{A}'_n(\mathcal{L})$. This sum is a sum over diagrams with at least two univalent vertices, according to Lemma 6.26. Recall Notation 7.29. Lemma 10.15 guarantees that

$$Z_{n,I}(\check{R}, L, (\omega(i))_{i \in I}) = \sum_{\Gamma \in \mathcal{D}_{n,I}^e(\mathcal{L})} \zeta_{\Gamma} I(R, L, \Gamma, (\omega(i))_{i \in I}) [\Gamma] \in \mathcal{A}_n(\mathcal{L})$$

is independent of the chosen $\omega(i)$ for any subset $I$ of $\underline{3n}$ with cardinality $(3n - 2)$. Observe

$$Z'_{n}(\check{R}, L, (\omega(i))_{i \in \underline{3n}}) = \frac{2((3n-2)!)}{(3n)!} \sum_{I \subset \underline{3n} : |I|=3n-2} Z_{n,I}(\check{R}, L, (\omega(i))_{i \in I})$$

since for a numbered graph $\Gamma$ of $\mathcal{D}_n^e(\mathcal{L})$, the coefficient of $I(R, L, \Gamma, (\omega(i))_{i \in \underline{3n}})$ is $\frac{(3n-|E(\Gamma)|)!}{(3n)!2^{|E(\Gamma)|}}$ [Γ] in the left-hand side, and

$$\frac{2((3n-2)!)}{(3n)!} \sum_{I \subset \underline{3n} : |I|=3n-2, j_E(E(\Gamma)) \subseteq I} \frac{(3n-2-|E(\Gamma)|)!}{(3n-2)!2^{|E(\Gamma)|}} [\Gamma] \\ = \frac{2((3n-2)!)}{(3n)!} \frac{(3n-|E(\Gamma)|)!}{2((3n-2-|E(\Gamma)|)!)} \frac{(3n-2-|E(\Gamma)|)!}{(3n-2)!2^{|E(\Gamma)|}} [\Gamma]$$

in the right-hand-side. Thus, Lemma 10.15 and Lemma 9.5 imply Theorem 7.40.

Lemma 10.15 also directly implies that

$$\sum_{\Gamma \in \mathcal{D}_{n,3n-2}^{e}(\mathcal{L})} \zeta_{\Gamma} I(R, L, \Gamma, (\omega(i))_{i \in \underline{3n-2}}) \check{p}([\Gamma])$$

is independent of the chosen $\omega(i)$, so it implies Theorem 7.42 as above. □

PROOF OF LEMMA 10.15: Let $L: \mathcal{L} \hookrightarrow \check{R}$ be a straight embedding with respect to $\tau$. It suffices to prove that $Z_{n,3n-2}(\check{R}, L, (\omega(i))_{i \in \underline{3n-2}})$ does not change when some $\omega(i)$ is changed to $\omega(i) + d\eta$ for some one-form $\eta$ on $C_2(R)$, which restricts to $\partial C_2(R)$ as $p_{\tau}^*(\eta_{S^2})$ for some one-form $\eta_{S^2}$ on $S^2$, as in Lemma 3.17. Assume that the forms $\omega(j)$ restrict to $\partial C_2(R)$ as $p_{\tau}^*(\omega_{S^2}(j))$. Set $\tilde{\omega}(i, 0) = \omega(i)$. Let $p_{C_2}: [0, 1] \times C_2(R) \to C_2(R)$ and $p_{S^2}: [0, 1] \times S^2 \to S^2$ denote the projections onto the second factor. Define the closed 2-form $\tilde{\omega}_{S^2}(i)$ on $[0, 1] \times S^2$ by

$$\tilde{\omega}_{S^2}(i) = p_{S^2}^*(\omega_{S^2}(i)) + d.tp_{S^2}^*(\eta_{S^2}),$$
where $t$ is the coordinate on $[0, 1]$. Define the closed 2-form $\tilde{\omega}(i)$ on $[0, 1] \times C_2(R)$ to be

$$ \tilde{\omega}(i) = p_{C_2}^*(\omega(i)) + d.tp_{C_2}^*(\eta). $$

For $j \in \underline{3n} \setminus \{i\}$, define $\tilde{\omega}_{S^2}(j) = p_{S^2}^*(\omega_{S^2}(j))$ and $\tilde{\omega}(j, t)$ denote the restriction of $\tilde{\omega}(j)$ to $\{t\} \times C_2(R)$. Thus, it suffices to prove that $Z_{n,3n-2}(1) = Z_{n,3n-2}(0)$, with

$$ Z_{n,3n-2}(t) = \sum_{\Gamma \in \mathcal{D}_{n,3n-2}^e(\mathcal{L})} \zeta_{\Gamma} I(R, L, \Gamma, (\tilde{\omega}(i,t))_{i \in \underline{3n-2}}) [\Gamma] \in \mathcal{A}_n(\mathcal{L}). $$

**Proposition 9.2** expresses $(Z_{n,3n-2}(1) - Z_{n,3n-2}(0))$ as a sum over numbered graphs $\Gamma$ equipped with a connected component $\Gamma_A$, with no univalent vertex, or whose univalent vertices form a nonempty set of consecutive vertices in $\Gamma$ on some component $S_j^1$ of $\mathcal{L}$. The faces for which $A$ has no univalent vertex do not contribute, as in the proof of Lemma 9.6. So we focus on the remaining faces.

Such a $F(A, L, \Gamma)$ may split according to the possible compatible linear orders of the univalent vertices of $\Gamma_A$, represented by lifts $\check{\Gamma}_A$ of $\Gamma_A$ on $\mathbb{R}$ as in Notation 8.15. We view $Z_{n,3n-2}(1) - Z_{n,3n-2}(0)$ as a sum over pairs $(\Gamma, \check{\Gamma}_A)$ of terms

$$ \zeta_{\Gamma} I(\Gamma, \check{\Gamma}_A) = \zeta_{\Gamma} \int_{[0,1] \times F(\check{\Gamma}_A, L, \Gamma)} \bigwedge_{e \in E(\Gamma)} p_e^* (\tilde{\omega}(j_E(e))) [\Gamma] $$

associated to the corresponding face components denoted by $F(\check{\Gamma}_A, L, \Gamma)$.

For a fixed numbered graph $\check{\Gamma}_A$ on $\mathbb{R}$ as above, we study the sum of the contributions $\zeta_{\Gamma} I(\check{\Gamma}, \check{\Gamma}_A)$ running over the graphs $\check{\Gamma}$ such that

* the graph $\check{\Gamma}_A$ is a subgraph of $\check{\Gamma}$ when $\mathbb{R}$ is identified with a part of $S_j^1$,

* the univalent vertices of $\check{\Gamma}_A$ are consecutive on $S_j^1$ in $\check{\Gamma}$, with respect to their linear order, and

* $\check{\Gamma} \setminus \check{\Gamma}_A$ is equal to a fixed $\Gamma \setminus \check{\Gamma}_A$, as above

Recall from Proposition 6.22 that $[\check{\Gamma}] = [\check{\Gamma}_A] \#_j [\Gamma \setminus \check{\Gamma}_A]$ in $\check{\mathcal{A}}(\mathcal{L})$ for any such pair $(\check{\Gamma}, \check{\Gamma}_A)$, provided that all the vertices of the graph $\check{\Gamma}$ inherit their orientations from a fixed vertex-orientation of $\Gamma$.

If $i \notin j_E(E(\check{\Gamma}_A))$, then the $I(\check{\Gamma}, \check{\Gamma}_A)$ vanish because all the $p_e^*(\tilde{\omega}(j_E(e)))$ for $e \in E(\check{\Gamma}_A)$ factor through the projection onto $F(\check{\Gamma}_A, L, \Gamma)$ whose dimension is $(2|E(\check{\Gamma}_A)| - 1)$.
If $i \in j_E(E(\check{\Gamma}_A))$, let $e(i)$ be the edge such that $j_E(e(i)) = i$. The sum of the contributions $\zeta \cdot I(., \check{\Gamma}_A)$ involving $\check{\Gamma}_A$ factors through

$$I = \int_{[0,1] \times \cup_{c(v) \in K_j} \check{S}(T_{c(v)}\check{R}, \vec{t}_{c(v)}; \check{\Gamma}_A)} p_{e(i)}^*(d.tp_{C_2}^*(\eta)) \wedge \bigwedge_{e \in E(\check{\Gamma}_A) \setminus e(i)} p_e^*(\tilde{\omega}(j_E(e))),$$

where $\vec{t}_{c(v)}$ denotes the unit tangent vector to $K_j$ at $c(v)$.

Recall that $\check{Q}(\check{\Gamma}_A)$ was defined in Section 10.3, together with natural maps

$$p_{e,S^2}: \check{Q}(\check{\Gamma}_A) \to S^2.$$

Let $p_{e,S^2}$ also denote $\mathbf{1}_{[0,1]} \times p_{e,S^2}: [0,1] \times \check{Q}(\check{\Gamma}_A) \to [0,1] \times S^2$. The form

$$p_{e(i)}^*(d.tp_{C_2}^*(\eta)) \wedge \bigwedge_{e \in E(\check{\Gamma}_A) \setminus e(i)} p_e^*(\tilde{\omega}(j_E(e)))$$

is the pull-back of the closed form

$$\Omega = p_{e(i),S^2}^*(d(t\eta_{S^2})) \wedge \bigwedge_{e \in E(\check{\Gamma}_A) \setminus e(i)} p_{e,S^2}^*(\tilde{\omega}_{S^2}(j_E(e)))$$

on $[0,1] \times \check{Q}(\check{\Gamma}_A)$ under the projection

$$[0,1] \times \cup_{c(v) \in K_j} \check{S}(T_{c(v)}\check{R}, \vec{t}_{c(v)}; \check{\Gamma}_A) \to [0,1] \times \check{Q}(\check{\Gamma}_A).$$

The image of this projection is the product by $[0,1]$ of the restriction of the bundle $\check{Q}(\check{\Gamma}_A)$ over $p_\tau(U^+K_j)$, and $I$ is the integral of $\Omega$ along this image. Compute the integral by integrating first along the fibers of $\check{Q}(\check{\Gamma}_A)$, next along $[0,1]$. Afterwards, the integral $I$ becomes the integral of a one-form along $p_\tau(U^+K_j) \subset S^2$. So it vanishes because $K_j$ is straight. $\square$

## 10.5 The general variation for homogeneous propagating forms

Set $\mathcal{D}^c(\mathbb{R}) = \cup_{n \in \mathbb{N}} \mathcal{D}_n^c(\mathbb{R})$, where $\mathcal{D}_n^c(\mathbb{R})$ is the set of degree $n$ connected $(3n-2)$-numbered Jacobi diagrams on $\mathbb{R}$ introduced in the beginning of Section 10.3. In this section, we write various sums over numbered diagrams, but all the edges of a diagram are equipped with the same propagating forms. So neither the set in which the edges are numbered nor its cardinality matters, provided that the cardinality is greater than the possible number of edges for a given degree. (See Proposition 7.25 and Remark 7.31.)
**Proposition 10.17.** Let $(\tilde{R}, \tau)$ be an asymptotic rational homology $\mathbb{R}^3$. Let $L = \bigsqcup_{j=1}^k K_j$ be an embedding of $\mathcal{L} = \bigsqcup_{j=1}^k S_j^1$ into $\tilde{R}$. Let $\tilde{\omega}(0)$ and $\tilde{\omega}(1)$ be two homogeneous propagating forms of $C_2(R)$. Let $\tilde{\omega}$ be a closed 2-form on $[0, 1] \times \partial C_2(R)$ whose restriction $\tilde{\omega}(t)$ to $\{t\} \times (\partial C_2(R) \setminus UB_R)$ is $p_\tau^*(\omega_{S^2})$ for any $t \in [0, 1]$, and whose restriction $\tilde{\omega}(t)$ to $\{t\} \times \partial C_2(R)$ coincides with the restriction to $\partial C_2(R)$ of the given $\tilde{\omega}(t)$ for $t \in \{0, 1\}$. For any $j \in \underline{k}$, define $I_j = \sum_{\Gamma_B \in \mathcal{D}^c(\mathbb{R})} \zeta_{\Gamma_B} I(\Gamma_B, K_j, \tilde{\omega})$, where

$$I(\Gamma_B, K_j, \tilde{\omega}) = \int_{u \in [0,1]} \int_{w \in K_j} \int_{\tilde{S}(T_w \tilde{R}, \tilde{t}_w; \Gamma_B)} \bigwedge_{e \in E(\Gamma_B)} p_e^*(\tilde{\omega}(u)) [\Gamma_B],$$

and $\tilde{t}_w$ denotes the unit tangent vector to $K_j$ at $w$. Define

$$z(\tilde{\omega}) = \sum_{n \in \mathbb{N}} z_n([0, 1] \times UB_R; \tilde{\omega})$$

as in Corollary 9.4. Then we have

$$Z(\check{R}, L, \tilde{\omega}(1)) = \left( \prod_{j=1}^{k} \exp(I_j) \right) \#_j Z(\check{R}, L, \tilde{\omega}(0)) \exp(z(\tilde{\omega})),$$

where $\#_j$ stands for the insertion on a diagram on $\mathbb{R}$ on the component $S_j^1$ of $\mathcal{L}$.

We will actually prove the following lemma.

**Lemma 10.18.** *Recall Notation 7.16. Under the assumptions of Proposition 10.17, we have*

$$\check{Z}(\check{R}, L, \tilde{\omega}(1)) = \left( \prod_{j=1}^{k} \exp(I_j) \right) \#_j \check{Z}(\check{R}, L, \tilde{\omega}(0)).$$

**Lemma 10.19.** *Lemma 10.18 implies Proposition 10.17.*

*PROOF:* When $L = \emptyset$, Proposition 10.17 follows from Corollary 9.4, Proposition 10.2, and Lemma 9.1, which ensures that there exists a closed 2-form $\tilde{\omega}$ on $[0, 1] \times C_2(R)$ that extends the 2-form $\tilde{\omega}$ of the statement. Conclude with Lemma 7.27. $\square$

Let us begin the proof of Lemma 10.18 with the proof of the following corollary of Proposition 9.2.
**Lemma 10.20.** Under the assumptions of Proposition 10.17, let $\tilde{\omega}$ be a closed 2-form on $[0, 1] \times C_2(R)$ which extends the 2-form $\tilde{\omega}$ of Proposition 10.17. For any $t \in [0, 1]$, let $\tilde{\omega}(t)$ denote the restriction to $\{t\} \times C_2(R)$ of $\tilde{\omega}$. Set

$$ \check{Z}(t) = (\check{Z}_n(R, L, \tilde{\omega}(t)))_{n \in \mathbb{N}}. $$

For $\Gamma_B \in \mathcal{D}^c(\mathbb{R})$, $u \in [0, 1]$, and $j \in \underline{k}$, set

$$ \eta(R, L, \Gamma_B, K_j, \tilde{\omega})(u) = \int_{w \in K_j} \int_{\check{S}(T_w \check{R}, \vec{t}_w; \Gamma_B)} \bigwedge_{e \in E(\Gamma_B)} p_e^*(\tilde{\omega}(u)) [\Gamma_B], $$

where $\vec{t}_w$ denotes the unit tangent vector to $K_j$ at $w$, and set

$$ \gamma_j(u) = \sum_{\Gamma_B \in \mathcal{D}^c(\mathbb{R})} \zeta_{\Gamma_B} \eta(R, L, \Gamma_B, K_j, \tilde{\omega})(u). $$

Then $\check{Z}(t)$ is differentiable, and we have

$$ \check{Z}'(t)dt = \left( \sum_{j=1}^{k} \gamma_j(t) \#_j \right) \check{Z}(t). $$

**PROOF OF LEMMA 10.20:** The variations of $\check{Z}_n(t)$ are given by Proposition 9.2, by sending the diagrams with components without univalent vertices to 0. They involve only faces $F(A, L, \Gamma)$ for which $\Gamma_A$ is a connected component of $\Gamma$, with univalent vertices on one component of $\mathcal{L}$. Again, such a face may split according to the possible compatible linear orders of the univalent vertices of $\Gamma_A$, represented by lifts $\check{\Gamma}_A$ of $\Gamma_A$ on $\mathbb{R}$, as in Notation 8.15. The corresponding face component is denoted by $F(\check{\Gamma}_A, L, \Gamma)$, and the corresponding integral is

$$ I(\Gamma, \check{\Gamma}_A) = \int_{[0,t] \times F(\check{\Gamma}_A, L, \Gamma)} \bigwedge_{e \in E(\Gamma)} p_e^*(\tilde{\omega}) [\Gamma]. $$

Proposition 9.2 implies

$$ \check{Z}_n(t) - \check{Z}_n(0) = \sum_{(\Gamma,A) : \Gamma \in \mathcal{D}_n^e(\mathcal{L}), A \subseteq V(\Gamma), |A| \ge 2,} \zeta_{\Gamma} I(\Gamma, \check{\Gamma}_A), $$
every component of $\Gamma$ has univalent vertices,
$\Gamma_A$ is a connected component of $\Gamma$,
the univalent vertices of $\Gamma_A$ are consecutive
on one component of $\Gamma$, and
$\check{\Gamma}_A$ is a compatible lift of $\Gamma_A$ on $\mathbb{R}$. $$
where the set of univalent vertices of $\check{\Gamma}_A$ is equipped with the unique linear order induced by $\Gamma$ if there are univalent vertices of $\Gamma \setminus \Gamma_A$ on the component of $\Gamma_A$, and with one of the linear orders compatible with $\Gamma$ otherwise.

This expression implies that $\check{Z}_n$ (valued in a finite-dimensional vector space) is differentiable. (For any smooth compact $d$-dimensional manifold $C$ and for any smooth $(d+1)$-form $\omega$ on $[0, 1] \times C$, the function $(t \mapsto \int_{[0,t] \times C} \omega)$ is differentiable.) Assume that the vertices of $\Gamma_A$ are on a component $K_j$ of $L(\bigsqcup_{j=1}^k S_j^1)$. The forms associated to edges of $\Gamma_A$ are integrated along $[0, 1] \times (\cup_{c(v) \in K_j} \check{S}(T_{c(v)}\check{R}, \vec{t}_{c(v)}; \check{\Gamma}_A))$, where $\vec{t}_{c(v)}$ denotes the unit tangent vector to $K_j$ at $c(v)$. So they do not depend on the configuration of $(V(\Gamma) \setminus A)$. The other forms are integrated along $\check{C}(R, L; \Gamma \setminus \Gamma_A)$ at $u \in [0, 1]$.

Group the contributions of the pairs $(\Gamma, \check{\Gamma}_A)$ with common $(\Gamma \setminus \Gamma_A, \check{\Gamma}_A)$ to view the global variation $(\check{Z}(t) - \check{Z}(0))$ as

$$ \sum_{j=1}^{k} \int_{0}^{t} \left( \sum_{\check{\Gamma}_A \in \mathcal{D}^c(\mathbb{R})} \zeta_{\check{\Gamma}_A} \eta(R, L, \check{\Gamma}_A, K_j, \tilde{\omega})(u) \#_j \right) \check{Z}(u). $$

Use Proposition 7.25 and Remark 7.26 to check that the coefficients are correct. So we get

$$ \check{Z}(t) - \check{Z}(0) = \int_{0}^{t} \left( \sum_{j=1}^{k} \gamma_{j}(u) \#_{j} \right) \check{Z}(u). $$

□

PROOF OF LEMMA 10.18: Set $I_j(t) = \int_0^t \gamma_j(u)$. We have $\check{Z}_0(t) = 1$. The equation

$$ \check{Z}'(t)dt = \left( \sum_{j=1}^{k} \gamma_j(t) \#_j \right) \check{Z}(t) $$

of Lemma 10.20 determines $\check{Z}(t)$ as a function of $\check{Z}(0)$ by induction on the degree, and we have $\check{Z}(t) = \prod_{j=1}^k \exp(I_j(t)) \#_j \check{Z}(0)$. □

Let us now apply Lemma 10.20 to study the variation of the quantity $\check{z}(R, L, \tau)$ of Corollary 10.4 when $\tau$ varies smoothly.

**Lemma 10.21.** Let $(\tau(t))_{t \in [0,1]}$ define a smooth homotopy of asymptotically standard parallelizations of $\check{R}$. We have

$$ \frac{\partial}{\partial t} \check{Z}(\check{R}, L, \tau(t)) = \left( \sum_{j=1}^{k} \frac{\partial}{\partial t} \left( 2 \int_{[0,t] \times U^+ + K_j} p_{\tau(.)}^*(\omega_{S^2}) \right) \alpha \#_j \right) \check{Z}(\check{R}, L, \tau(t)). $$
PROOF: Fix a homogeneous propagating form $\omega$ of $(C_2(R), \tau(0))$ and a form $\tilde{\omega}$ on $[0, 1] \times C_2(R)$ such that $\tilde{\omega}(t)$ is a homogeneous propagating form $\omega$ of $(C_2(R), \tau(t))$ for all $t \in [0, 1]$ as in Lemma 9.1. Lemma 10.20 ensures

$$ \frac{\partial}{\partial t} \check{Z}(\check{R}, L, \tau(t)) dt = \left( \sum_{j=1}^{k} \gamma_j(t) \#_j \right) \check{Z}(\check{R}, L, \tau(t)), $$

with

$$ \gamma_j(u) = \sum_{\Gamma_B \in \mathcal{D}^c(\mathbb{R})} \zeta_{\Gamma_B} \eta(R, L, \Gamma_B, K_j, \tilde{\omega})(u), $$

and

$$ \eta(R, L, \Gamma_B, K_j, \tilde{\omega})(u) = \int_{w \in K_j} \int_{\check{S}(T_w \check{R}, \vec{t}_w; \Gamma_B)} \bigwedge_{e \in E(\Gamma_B)} p_e^*(p_{\tau(u)}^*(\omega_{S^2})) [\Gamma_B]. $$

The restriction of $p_{\tau(.)}$ from $[0, 1] \times U^+ K_j$ to $S^2$ induces a map

$$ p_{a,\tau,\Gamma_B} : [0,1] \times \cup_{w \in K_j} \check{S}(T_w \check{R}, \vec{t}_w; \Gamma_B) \to \check{Q}(\Gamma_B) $$

over $(p_{\tau(.)}: [0, 1] \times U^+ K_j \to S^2)$, which restricts to the fibers as the identity map, for any $\Gamma_B \in \mathcal{D}^c(\mathbb{R})$. (Recall the definition of $\check{Q}(\Gamma_B)$ from the beginning of Section 10.3.) We have

$$ \int_0^1 \eta(R, L, \Gamma_B, K_j, \tilde{\omega})(u) = \int_{\mathrm{Im}(p_{a,\tau,\Gamma_B})} \left( \bigwedge_{e \in E(\Gamma_B)} p_{e,S^2}^*(\omega_{S^2}) \right) [\Gamma_B]. $$

Integrating $(\bigwedge_{e \in E(\Gamma_B)} p_{e,S^2}^*(\omega_{S^2})) [\Gamma_B]$ along the fiber in $\check{Q}(\Gamma_B)$ yields a two-form on $S^2$. This two-form is homogeneous because everything is. Thus, it may be expressed as $2\alpha(\Gamma_B)\omega_{S^2} [\Gamma_B]$, where $\alpha(\Gamma_B) \in \mathbb{R}$ and

$$ \sum_{\Gamma_B \in \mathcal{D}^c(\mathbb{R})} \zeta_{\Gamma_B} \alpha(\Gamma_B) [\Gamma_B] = \alpha. $$

So we get

$$ \int_0^t \eta(R, L, \Gamma_B, K_j, \tilde{\omega})(u) = 2\alpha(\Gamma_B) \int_{[0,t] \times U^+ K_j} p_{\tau(.)}^*(\omega_{S^2}) [\Gamma_B], $$

$$ \text{and } \gamma_j(t) = 2 \frac{\partial}{\partial t} \left( \int_{[0,t] \times U^+ K_j} p_{\tau(.)}^*(\omega_{S^2}) \right) \alpha dt. \quad \square $$
**Corollary 10.22.** *The product*

$$ \prod_{j=1}^{k} \exp(-I_{\theta}(K_j, \tau(t))\alpha) \#_j \check{Z}(\check{R}, L, \tau(t)) $$

does not change when $\tau$ varies by a smooth homotopy.

**PROOF:** With the notation of Lemma 7.15, we have

$$ \check{Z}_1(\check{R}, L, \tau(t)) = \frac{1}{2} \sum_{j=1}^{k} I_{\theta}(K_j, \tau(t)) [\stackrel{\leftarrow}{\bullet}^{S_j^1}] $$

Therefore, Lemma 10.21 and Proposition 10.13 imply

$$ \frac{\partial}{\partial t} I_{\theta}(K_j, \tau(t)) = 2 \frac{\partial}{\partial t} \int_{[0,t] \times U + K_j} p_{\tau(.)}^{*}(\omega_{S^2}). $$

(The reader can also check it directly as an exercise.) So Lemma 10.21 implies

$$ \frac{\partial}{\partial t} \check{Z}(\check{R}, L, \tau(t)) = \sum_{j=1}^{k} \left( \frac{\partial}{\partial t} I_{\theta}(K_j, \tau(t)) \alpha_{\#_j} \right) \check{Z}(\check{R}, L, \tau(t)). $$

Therefore, the derivative of

$$ \prod_{j=1}^{k} \exp(-I_{\theta}(K_j, \tau(t))\alpha) \#_j \check{Z}(\check{R}, L, \tau(t)) $$

with respect to $t$ vanishes.

<figure><img src="image_86.png" /></figure>

**PROOF OF THEOREM 7.20:** According to the naturality of Proposition 10.1, Lemma 9.5, Proposition 10.7, Corollary 10.9, and Proposition 10.13, it suffices to prove that

$$ \prod_{j=1}^{k} \left( \exp(-I_{\theta}(K_j, \tau)\alpha) \right) \#_j \check{Z}(\check{R}, L, \tau) \in \check{\mathcal{A}}(\mathcal{L}) $$

is independent of the homotopy class of parallelization $\tau$.

When $\tau$ changes in a ball that does not meet the link, the forms can be changed only in the neighborhoods of the unit tangent bundle to this ball. Apply Proposition 10.17 again to $\check{Z}$, where the $p_e^*(\tilde{\omega}(u))$ are independent of $u$ over $K_j$. So we get that

$$ \prod_{j=1}^{k} \left( \exp(-I_{\theta}(K_j, \tau)\alpha) \right) \#_j \check{Z}(\check{R}, L, \tau) \in \check{\mathcal{A}}(\mathcal{L}) $$
is invariant under the natural action of $\pi_3(SO(3))$ on the homotopy classes of parallelizations, thanks to Corollary 10.22.

Now examine the effect of the twist of the parallelization by a map $g: (B_R, 1) \to (SO(3), 1)$. Without loss of generality, assume that $p_\tau(U^+K_j) = v$ for some $v \in S^2$ and $g$ maps $K_j$ to rotations with axis $v$ for any $j \in \underline{k}$. We want to compare $\check{Z}(\check{R}, L, \tau \circ \psi_\mathbb{R}(g))$ with $\check{Z}(\check{R}, L, \tau)$. There exists a closed form $\omega$ on $[0, 1] \times UB_R$ equal to $p_\tau^*(\omega_{S^2})$ on $\partial([0, 1] \times UB_R) \setminus (\{1\} \times UB_R)$ and equal to $p_{\tau \circ \psi_\mathbb{R}(g)}^*(\omega_{S^2})$ on $\{1\} \times UB_R$. Extend this form to a closed form $\Omega$ on $[0, 1] \times C_2(R)$, which restricts to $[0, 1] \times (\partial C_2(R) \setminus UB_R)$ as $p_\tau^*(\omega_{S^2})$, as in Lemma 9.1. Let $\Omega(t)$ denote the restriction of $\Omega$ to $\{t\} \times C_2(R)$. According to Proposition 10.17, we have

$$ \check{Z}(\check{R}, L, \tau \circ \psi_\mathbb{R}(g)) = \prod_{j=1}^{k} (\exp(I_j)\#_j) \check{Z}(\check{R}, L, \tau), $$

where $I_j = \int_0^1 \gamma_j(u)$, with $\gamma_j(t) = \sum_{\Gamma_B \in \mathcal{D}^c(\mathbb{R})} \zeta_{\Gamma_B} \eta(R, L, \Gamma_B, K_j, \Omega)(t)$ and

$$ \eta(R, L, \Gamma_B, K_j, \Omega)(t) = \int_{w \in K_j} \int_{\check{S}(T_w \check{R}, \vec{t}_w; \Gamma_B)} \left( \bigwedge_{e \in E(\Gamma_B)} p_e^*(\Omega(t)) \right) [\Gamma_B]. $$

It suffices to prove $I_j = (I_\theta(K_j, \tau \circ \psi_\mathbb{R}(g)) - I_\theta(K_j, \tau))\alpha$. Proposition 10.17 implies that the degree one part $I_{1,j}$ of $I_j$ is

$$ \begin{aligned} I_{1,j} &= \check{Z}_1(\check{R}, K_j, \tau \circ \psi_\mathbb{R}(g)) - \check{Z}_1(\check{R}, K_j, \tau) \\ &= \frac{1}{2} (I_\theta(K_j, \tau \circ \psi_\mathbb{R}(g)) - I_\theta(K_j, \tau)) [\zeta]. \end{aligned} $$

Let $\tau^\psi(g^{-1}) : UB_R \to UB_R$ denote the map induced by $\tau \circ \psi_\mathbb{R}(g^{-1}) \circ \tau^{-1}$. Recall $p_\tau = p_{S^2} \circ \tau^{-1}$. So we have

$$ p_{\tau \circ \psi_\mathbb{R}(g)} = p_{S^2} \circ \psi_\mathbb{R}(g^{-1}) \circ \tau^{-1} = p_\tau \circ \tau^\psi(g^{-1}). $$

Let $(-1): [1, 2] \to [0, 1]$ map $x$ to $x-1$. Set $\tau^\psi(g^{-1}) = ((-1) \times \tau^\psi(g^{-1}))$. Extend $\Omega$ over $[0, 2] \times C_2(R)$ so that $\Omega$ restricts to $[1, 2] \times UB_R$ as $\tau^\psi(g^{-1})^*(\Omega)$. For any $\Gamma_B$, the map $\tau^\psi(g^{-1})$ induces an orientation-preserving diffeomorphism

$$ \tau^\psi(g^{-1})_*: [1, 2] \times \cup_{w \in K_j} \check{S}(T_w \check{R}, \vec{t}_w; \Gamma_B) \to [0, 1] \times \cup_{w \in K_j} \check{S}(T_w \check{R}, \vec{t}_w; \Gamma_B) $$

such that $p_e \circ \tau^\psi(g^{-1})_* = \tau^\psi(g^{-1}) \circ p_e$ for any edge $e$ of $\Gamma_B$. Using these diffeomorphisms $\tau^\psi(g^{-1})_*$ to pull back $(\bigwedge_{e \in E(\Gamma_B)} p_e^*(\Omega(t)))$ proves $\gamma_j(t+1) = \gamma_j(t)$. In particular, we get

$$ I_j(2) = \int_0^2 \gamma_j(u) = 2I_j. $$
Set $\tilde{Z}(2) = \tilde{Z}(R, L, \tau \circ \psi_{\mathbb{R}}(g)^2)$. We have

$$ \tilde{Z}(2) = \prod_{j=1}^{k} \exp \left( \left( I_{\theta}(K_j, \tau \circ \psi_{\mathbb{R}}(g)^2) - I_{\theta}(K_j, \tau) \right) \alpha \right) \#_j \tilde{Z}(\tilde{R}, L, \tau), $$

since $g^2$ is homotopic to the trivial map outside a ball (see Lemma 5.2, 2). Proposition 10.17 implies

$$ \tilde{Z}(2) = \prod_{j=1}^{k} (\exp(2I_j)\#_j) \tilde{Z}(\tilde{R}, L, \tau). $$

By induction on the degree, we get

$$ 2I_j = (I_{\theta}(K_j, \tau \circ \psi_{\mathbb{R}}(g)^2) - I_{\theta}(K_j, \tau))\alpha. $$

The degree one part of this equality implies

$$ 2(I_{\theta}(K_j, \tau \circ \psi_{\mathbb{R}}(g)) - I_{\theta}(K_j, \tau)) = I_{\theta}(K_j, \tau \circ \psi_{\mathbb{R}}(g)^2) - I_{\theta}(K_j, \tau). $$

We get $I_j = (I_{\theta}(K_j, \tau \circ \psi_{\mathbb{R}}(g)) - I_{\theta}(K_j, \tau)) \alpha$, as desired. $\square$

<figure><img src="image_87.png" /></figure>

## 10.6 Some more properties of Z

When $\tilde{R} = \mathbb{R}^3$, then $\mathcal{Z}(S^3, L) = \tilde{Z}(S^3, L)$ is the configuration space invariant studied by Daniel Altschüler, Laurent Freidel [AF97], Dylan Thurston [Thu99], Sylvain Poirier [Poi02], and others, after the work of many people including Edward Witten [Wit89], Enore Guadagnini, Maurizio Martellini, Mihail Mintchev [GMM90], Maxim Kontsevich [Kon94, Kon93], Raoul Bott and Clifford Taubes [BT94], Dror Bar-Natan [BN95b]...

**Reversing a link component orientation.** The following proposition is obvious from the definition of Z.

**Proposition 10.23.** Let $L: \bigsqcup_{j=1}^k S_j^1 \to R$ be a link in a $\mathbb{Q}$-sphere $R$. For a Jacobi diagram $\Gamma$ on $\bigsqcup_{j=1}^k S_j^1$, let $U_j(\Gamma)$ denote the set of univalent vertices of $\Gamma$ mapped to $S_j^1$. This set is cyclically ordered by $S_j^1$. When the orientation of the component $L(S_j^1)$ is reversed, $\mathcal{Z}(L)$ is modified by reversing the circle $S_j^1$ (that is reversing the cyclic order of $U_j(\Gamma)$) in classes $[\Gamma]$ of diagrams $\Gamma$ on $\bigsqcup_{j=1}^k S_j^1$ and multiplying them by $(-1)^{|U_j(\Gamma)|}$ in $\mathcal{A}(\bigsqcup_{j=1}^k S_j^1)$.

In other words, we can forget the orientation of the link $L$ and view $\mathcal{Z}(L)$ as valued in $\mathcal{A}(\bigsqcup_{j=1}^k S_j^1)$, where the $S_j^1$ are not oriented, as in Definitions 6.13 and 6.16.
**Remark 10.24.** The orientation of a component $L(S_j^1)$ is used in two ways. It defines a cyclic order on $U_j(\Gamma)$, and it defines the orientation of the vertices of $U_j(\Gamma)$ as in Definition 6.13. The local orientation of $S_j^1$ near the image of a vertex orients the corresponding local factor of the configuration space. The cyclic order is encoded in the isotopy class of the injection of $U_j$ into the domain $S_j^1$.

**Numbering of the link components.** The following proposition is obvious from the definition of $\mathcal{Z}$.

**Proposition 10.25.** When the numbering of the components of $L$ is changed, $\mathcal{Z}(L)$ is modified by the corresponding change of numbering of the circles $S_j^1$ in diagram classes of $\mathcal{A}(\bigsqcup_{j=1}^k S_j^1)$.

For a link $L: \mathcal{L}(L) \to R$ in a $\mathbb{Q}$-sphere $R$, $\mathcal{Z}(R, L)$ is valued in $\mathcal{A}(\mathcal{L}(L))$. The one-manifold $\mathcal{L}(L)$ is a disjoint union of oriented circles, which have been numbered so far. However, the numbers may be changed to any decoration that marks the component.

This gives sense to the statement of the following theorem.

**Connected sums.**

**Theorem 10.26.** For any two links $L_1$ and $L_2$ in rational homology spheres $R_1$ and $R_2$, we have

$$ \mathcal{Z}(R_1\#R_2, L_1 \sqcup L_2) = \mathcal{Z}(R_1, L_1)\mathcal{Z}(R_2, L_2). $$

We prove a generalization of Theorem 10.26 in Section 17.2. See Theorem 17.9 in particular. See also Section 13.3. The proof given in Section 17.2 can be read without reading the intermediate chapters. Theorem 10.26 and Corollary 10.11 yield the following corollary.

**Corollary 10.27.** For any two rational homology spheres $R_1$ and $R_2$, we have

$$ \Theta(R_1\#R_2) = \Theta(R_1) + \Theta(R_2). $$

**Reversing the orientation of the ambient space.**

**Lemma 10.28.** Under the assumptions of Lemma 5.14, we have

$$ Z_n(-\check{R}, L, \tau) = (-1)^n Z_n(\check{R}, L, \tau). $$
**PROOF:** If $\omega$ is a homogeneous propagating form of $(C_2(R), \tau)$, then $(-\omega)$ is a homogeneous propagating form of $(C_2(-R), \bar{\tau})$. Let $\Gamma$ be a degree $n$ numbered Jacobi diagram. When the orientation of $R$ is reversed, the orientation of $\check{C}(R, L; \Gamma)$ is reversed if and only if $|T(\Gamma)|$ is odd. Thus, the integrals will be multiplied by $(-1)^{|E(\Gamma)|+|T(\Gamma)|}$. We have $2|E(\Gamma)| = 3|T(\Gamma)| + |U(\Gamma)|$, and hence $2n = 2(|E(\Gamma)| - |T(\Gamma)|)$. $\square$

**Theorem 10.29.** For any link $L$ in a rational homology sphere $R$, we have

$$ \mathcal{Z}_n(-R, L) = (-1)^n \mathcal{Z}_n(R, L). $$

**PROOF:** Theorem 7.20 implies

$$ \mathcal{Z}(-R, L) = \exp\left(-\frac{1}{4}p_1(\bar{\tau})\beta\right) \prod_{j=1}^{k} \left(\exp(-I_{\theta}(K_j, \bar{\tau})\alpha)\#_j\right) Z(-\check{R}, L, \bar{\tau}). $$

Lemma 10.28 implies $I_{\theta}(K_j, \bar{\tau}) = -I_{\theta}(K_j, \tau)$, and Lemma 5.14 yields $p_1(\bar{\tau}) = -p_1(\tau)$. So Lemma 10.28 implies the result since $\alpha$ and $\beta$ vanish in even degrees. $\square$
<figure><img src="image_88.png" /></figure>
# Chapter 11

## Rationality

In this chapter, we give equivalent definitions of $\mathcal{Z}$ based on algebraic intersections of propagating chains, and we prove that $\mathcal{Z}$ and the anomalies $\alpha$ and $\beta$ are rational.

### 11.1 From integrals to algebraic intersections

In order to warm up, we first prove the following rationality result, which is due to Sylvain Poirier [Poi02] and Dylan Thurston [Thu99], independently.

**Proposition 11.1.** The anomaly $\alpha$ of Section 10.3 is rational, i.e., it belongs to $\tilde{\mathcal{A}}(\mathbb{R}; \mathbb{Q})$. For any link $L: \mathcal{L} \to \mathbb{R}^3$, the value $\mathcal{Z}(S^3, L)$ is also rational. It belongs to $\mathcal{A}(\mathcal{L}; \mathbb{Q})$.

**PROOF:** Let us fix $n$ and prove that $\alpha_n$ is in $\tilde{\mathcal{A}}_n(\mathbb{R}; \mathbb{Q})$. For any degree $n$ numbered Jacobi diagram $\tilde{\Gamma}$ on $\mathbb{R}$, define the smooth map

$$g(\tilde{\Gamma}): Q(\tilde{\Gamma}) \times (S^2)^{\frac{3n-2}{j_E(E(\tilde{\Gamma}))}} \to (S^2)^{3n-2}$$

as the product $(\prod_{e \in E(\tilde{\Gamma})} p_{e,S^2}) \times 1((S^2)^{\frac{3n-2}{j_E(E(\tilde{\Gamma}))}})$. Note that a regular value of $g(\tilde{\Gamma})$ is not in the image of $\partial Q(\tilde{\Gamma}) \times (S^2)^{\frac{3n-2}{j_E(E(\tilde{\Gamma}))}}$. According to the Morse–Sard theorem 1.4, the set of regular values of $g(\tilde{\Gamma})$ is dense. Since $Q(\tilde{\Gamma})$ is compact, so are $\partial Q(\tilde{\Gamma})$, the boundary of the domain of $g(\tilde{\Gamma})$, and the subset of the domain of $g(\tilde{\Gamma})$ consisting of the points at which the derivative of $g(\tilde{\Gamma})$ is not surjective. Therefore, the set of regular values of $g(\tilde{\Gamma})$ is open. Thus, the finite intersection over all the $\tilde{\Gamma} \in \mathcal{D}_n^c(\mathbb{R})$ of the sets of regular values of the $g(\tilde{\Gamma})$ is also open and dense. Let $\prod_{i=1}^{3n-2} B(x_i)$ be a product of open balls of $S^2$ that is in this intersection. Then for any $\tilde{\Gamma} \in \mathcal{D}_n^c(\mathbb{R})$, the local degree of $g(\tilde{\Gamma})$ (which is an integer) is constant over
$\prod_{i=1}^{3n-2} B(X_i)$. In particular, if $\omega(i, S^2)$ is a volume-one form of $S^2$ that is supported on $B(X_i)$ for each $i \in \underline{3n-2}$, then $I(\check{\Gamma}, o(\check{\Gamma}), \omega(i, S^2))$, which is nothing but this integral local degree, is an integer for any $\check{\Gamma}$ in $D_n^c(\mathbb{R})$. Thus $\alpha_n$, which is defined in Proposition 10.13, is in $\check{A}_n(\mathbb{R}; \mathbb{Q})$.

For a fixed $n$ and a given $k$-component link $L$ of $S^3$, there exists a similar product $\prod_{i=1}^{3n} B_L(Y_i)$ of open balls of $S^2$ consisting of points of $(S^2)^{3n}$ that are regular values for all maps

$$ \left( \prod_{e \in E(\Gamma)} p_{e,S^2} \right) \times \mathbf{1} \left( (S^2)^{\underline{3n} \setminus j_E(E(\Gamma))} \right) : C(S^3, L; \Gamma) \times (S^2)^{\underline{3n} \setminus j_E(E(\Gamma))} \to (S^2)^{3n} $$

associated to Jacobi diagrams $\Gamma$ of $D_n^e(\mathcal{L})$. Then if $\omega(i, S^2)$ is a volume-one form of $S^2$ that is supported on $B_L(Y_i)$ for each $i \in \underline{3n}$,

$$ I \left( S^3, L, \Gamma, \left( p_{S^2}^* (\omega(i, S^2)) \right) \right) $$

is an integer for every $\Gamma$ of $D_n^e(\mathcal{L})$.

If the link is straight, then Theorem 7.40 implies that $Z_n(\mathbb{R}^3, L, \tau_s)$ is rational. Thus, $Z(\mathbb{R}^3, L, \tau_s)$ is rational for any straight link $L$ of $\mathbb{R}^3$. In particular $I_\theta(K, \tau_s)$ is rational for any component $K$ of a straight link $L$, and Theorem 7.20 together with the rationality of $\alpha$ implies that $Z(S^3, L)$ is rational. $\square$

With the notation of the above proof, the $P(i) = p_{S^2}^{-1}(y_i) \subset C_2(S^3)$ for $y_i \in B_L(Y_i)$ are propagating chains such that, for any $\Gamma$ of $D_n^e(\mathcal{L})$, the intersection over $E(\Gamma)$ of the $p_e^{-1}(P(j_E(e)))$ in $C(S^3, L; \Gamma)$ is transverse. The integral $I(S^3, L, \Gamma, (p_{S^2}^*(\omega(i, S^2))))$ is nothing but their algebraic intersection.

We will use Version 7.40 of Theorem 7.20 to replace the configuration space integrals with algebraic intersections in configuration spaces and thus prove the rationality of $Z^s$ for straight links in any rational homology sphere as follows.

**Definition 11.2.** A smooth map $f: B \to A$ is *transverse* to a submanifold $C$ of $A$ along a subset $K$ of $B$ if

$$ T_{f(x)}A = T_x f(T_x B) + T_{f(x)}C $$

for any point $x$ of $K \cap f^{-1}(C)$. When $A$ or $B$ have ridges, we furthermore require this equality to hold when $A$ or $B$ are replaced by all their open faces (of any dimension).

A smooth map $f: B \to A$ is *transverse* to a submanifold $C$ of $A$ if it is transverse to $C$ along $B$. Say that a smooth map $f: B \to A$ is *transverse* to
a rational chain $C$ of $A$, which is a multiple of a union of compact smooth embedded submanifolds with boundaries and ridges $\cup_{k \in J} C_k$, if $f$ is transverse to $C_k$ for any $k \in J$.

A rational simplicial chain, which is a rational combination of simplices in a triangulated smooth manifold, is an example of what we call a rational chain. Rational multiples of compact immersion images provide other examples of chains. An immersion image will be represented as a union of embedded manifolds by decomposing the domain as a union of compact manifolds with boundaries and ridges glued along their boundaries.

Recall from Notation 7.29 that $\mathcal{D}_{k,3n}^e(\mathcal{L})$ denotes the set of $3n$-numbered degree $k$ Jacobi diagrams with support $\mathcal{L}$ without looped edges. Let $\mathcal{D}_{3n}^e(\mathcal{L}) = \cup_{k \in \mathbb{N}} \mathcal{D}_{k,3n}^e(\mathcal{L})$. Note that $\mathcal{D}_n^e(\mathcal{L}) = \mathcal{D}_{n,3n}^e(\mathcal{L})$ but $\underline{3n}$-numbered Jacobi diagrams may have a degree different from $n$.

**Definition 11.3.** Say that a family $(P(i))_{i \in \underline{3n}}$ of propagating chains of $(C_2(R), \tau)$ is in general $3n$ position with respect to a link $L: \mathcal{L} \to \tilde{R}$ if for any $\Gamma \in \mathcal{D}_{3n}^e(\mathcal{L})$ and for any subset $E$ of $E(\Gamma)$, the map

$$p(\Gamma, E) = \prod_{e \in E} p_e : C(R, L; \Gamma) \to (C_2(R))^{j_E(E)}$$

is transverse to $\prod_{e \in E} P(j_E(e))$.

For such a family $(P(i))_{i \in \underline{3n}}$ in general $3n$ position, the intersection

$$\cap_{e \in E(\Gamma)} p_e^{-1}(P(j_E(e)))$$

consists of a finite number of points $x$, which sit in the interior of $C(R, L; \Gamma)$, and, for each such $x$, the following conditions are satisfied.

* For every edge $e \in E(\Gamma)$, $p_e(x)$ meets the union of smooth embedded 4-manifolds with boundaries that constitute $P(j_E(e))$ in the interior of finitely many of these manifolds. The family $(\Delta_{j_E(e),i})_{i \in J(e,x)}$ of met manifolds is indexed by a finite set $J(e, x)$.<fnref n="1" />

* For every map $i: E(\Gamma) \to \cup_{e \in E(\Gamma)} J(e, x)$ such that $i(e) \in J(e, x)$, the local maps, from small open neighborhoods of $x$ in $C(R, L; \Gamma)$ to the product over $E(\Gamma)$ of the fibers of the locally trivialized normal bundles to the $\Delta_{j_E(e),i(e)}$, are local diffeomorphisms.

We will prove the following lemma in Section 11.3.

<footnote n="1">¹The $\Delta_{j_E(e),i}$ are smooth embedded 4-simplices when $P(j_E(e))$ is a simplicial chain.</footnote>