Let $\eta_{[-1,1]}$ be a one-form with compact support in $]-1, 1[$ such that

$$ \int_{[-1,1]} \eta_{[-1,1]} = 1. $$

Let $a_j^i \times [-1, 1]$ be a tubular neighborhood of $a_j^i$ in $\partial A^{(i)}$. Let $\eta(a_j^i)$ be a closed one-form on $A^{(i)}$ such that the support of $\eta(a_j^i)$ intersects $[-4, 0] \times \partial A^{(i)}$ inside $[-4, 0] \times (a_j^i \times [-1, 1])$, where $\eta(a_j^i)$ can be written as

$$ \eta(a_j^i) = p^*_{[-1,1]}(\eta_{[-1,1]}), $$

with the projection $p_{[-1,1]}: [-4, 0] \times (a_j^i \times [-1, 1]) \to [-1, 1]$ to the $[-1, 1]$ factor. Let $\eta(a_j^i)$ also denote a closed one-form on $A^{(i)'}$ which can be written in the same way on $[-4, 0] \times \partial A^{(i)}$. Note that the forms $\eta(a_j^i)$ on $A^{(i)}$ and $A^{(i)'}$ induce a closed one-form on $(A^{(i)} \cup_{\partial A^{(i)}} -A^{(i)'})$ that restrict to the previous ones on $A^{(i)}$ and $(-A^{(i)'}$). This one-form is also denoted by $\eta(a_j^i)$. The form $\eta(a_j^i)$ on $(A^{(i)} \cup_{\partial A^{(i)}} -A^{(i)'}$ is Poincaré dual to the homology class $\partial_{MV}^{-1}(a_j^i)$ in $(A^{(i)} \cup_{\partial A^{(i)}} -A^{(i)'})$, with the notation introduced before Theorem 18.6. Define the part $A_I^{(i)}$ of $\mathcal{C}_I$ to be $A_I^{(i)} = A^{(i)}$ if $i \notin I$ and $A_I^{(i)} = A^{(i)'}$ if $i \in I$.

The following proposition is the key to the proof of Theorem 18.6. Its proof is more complicated than I expected. We give it in Chapter 20.

**Proposition 18.33.** There exist homogeneous antisymmetric propagating forms $\omega_I$ of $(C_2(R_I), \tau_I)$ with the following properties.

* For any subsets $I$ and $J$ of $\underline{x}$, $\omega_I$ and $\omega_J$ coincide on

$$ C_2\left(\left(R \setminus \cup_{i \in I \cup J} \mathrm{Int}(A^{(i)})\right) \cup_{i \in I \cap J} A^{(i)'}\right), $$

* For any $(i, k) \in \underline{x}^2$ such that $i \neq k$, $\omega_I$ can be written as follows on $A_I^{(i)} \times A_I^{(k)}$:

$$ \omega_I = \sum_{\substack{j=1, \dots, g_i \\ \ell=1, \dots, g_k}} lk(z_j^i, z_\ell^k) p_{A_I^{(i)}}^*(\eta(a_j^i)) \wedge p_{A_I^{(k)}}^*(\eta(a_\ell^k)), $$

where $p_{A_I^{(i)}}: A_I^{(i)} \times A_I^{(k)} \to A_I^{(i)}$ and $p_{A_I^{(k)}}: A_I^{(i)} \times A_I^{(k)} \to A_I^{(k)}$ again denote the natural projections onto the factor corresponding to the subscript.

Recall Notation 18.5. Let $\Gamma$ be an oriented Jacobi diagram without univalent vertices. When $x$ is even and when $G = \bigsqcup_{i \in \underline{x}} T(\mathcal{I}_{A^{(i)} A^{(i)'})}$, define

$$ \langle\langle G \rangle\rangle_{\Gamma} = \sum_{p \in P(G) : \Gamma_p \text{ isomorphic to } \Gamma} [\ell(p) \Gamma_p], $$
where the sum runs over the $p$ such that $\Gamma_p$ is isomorphic to $\Gamma$ as a nonoriented trivalent graph.

Assuming Proposition 18.33, one can prove the following lemma.

**Lemma 18.34.** Let $\omega_I$ be forms as in Proposition 18.33. Let $\Gamma$ be an oriented Jacobi diagram on $\mathcal{L}$. If $\Gamma$ has less than $x$ trivalent vertices, then we have

$$ \sum_{I \subseteq \underline{x}} (-1)^{|I|} I(R_I, L, \Gamma, o(\Gamma), (\omega_I)) = 0. $$

If $\Gamma$ is a trivalent Jacobi diagram with $x$ vertices, then we have

$$ \sum_{I \subseteq \underline{x}} (-1)^{|I|} I(R_I, L, \Gamma, (\omega_I)) [\Gamma] = |\mathrm{Aut}(\Gamma)| \left\langle \left\langle \bigsqcup_{i \in \underline{x}} T(\mathcal{I}_{A^{(i)}A^{(i')}}) \right\rangle \right\rangle_{\Gamma}. $$

**PROOF:** Let $n \in \mathbb{N}$. Let $\Gamma$ be an oriented degree $n$ Jacobi diagram on $\mathcal{L}$. Let us compute

$$ \Delta = \sum_{I \subseteq \underline{x}} (-1)^{|I|} I(R_I, L, \Gamma, o(\Gamma), (\omega_I)). $$

Number the vertices of $\Gamma$ in $2n$ arbitrarily. So the open configuration space $\check{C}(\check{R}_I, L; \Gamma)$ becomes a submanifold of $\check{R}_I^{2n}$. The order of the vertices or- ders the oriented local factors (some of which are tangle components) of $\check{C}(\check{R}_I, L; \Gamma)$. Thus, it orients $\check{C}(\check{R}_I, L; \Gamma)$. Orient the edges of $\Gamma$ so that the edge-orientation of $H(\Gamma)$ and the vertex-orientation of $\Gamma$ induce the above orientation of $\check{C}(\check{R}_I, L; \Gamma)$ as in Lemma 7.1.

Let $i \in \underline{x}$. The forms $\bigwedge_{e \in E(\Gamma)} p_e^*(\omega_I)$ over

$$ \check{C}(\check{R}_I, L; \Gamma) \cap (\check{R}_I \setminus A_I^{(i)})^{2n} $$

are identical for $I = K$ and $I = K \cup \{i\}$ for any $K \subseteq \underline{x} \setminus \{i\}$. Since their integrals enter the sum $\Delta$ with opposite signs, they cancel each other. This argument allows us to get rid of the contributions of the integrals over the $\check{C}(\check{R}_I, L; \Gamma) \cap (\check{R}_I \setminus A_I^{(1)})^{2n}$ for any $I \subseteq \underline{x}$. The contributions over the

$$ (\check{C}(\check{R}_I, L; \Gamma) \setminus (\check{C}(\check{R}_I, L; \Gamma) \cap (\check{R}_I \setminus A_I^{(1)})^{2n}) \cap (\check{R}_I \setminus A_I^{(2)})^{2n} $$

cancel in the same way. Iterating, we get rid of the contributions of the integrals over the $\check{C}(\check{R}_I, L; \Gamma) \cap (\check{R}_I \setminus A_I^{(i)})^{2n}$ for any $i \in \underline{x}$, and for any $I \subseteq \underline{x}$. Thus, we are left with the contributions of the integrals over the subsets $\check{Q}_I$ of

$$ \check{C}(\check{R}_I, L; \Gamma) \subset \check{R}_I^{2n} $$
with the following property:

For any $i \in \underline{x}$, any element of $Q_I$ is sent to $A_I^{(i)}$ under at least one of the (2n) projections onto $\tilde{R}_I$. These subsets $Q_I$ are empty if $2n < x$. Thus, the lemma is proved when $\Gamma$ has less than $x$ trivalent vertices.

Assume $\Gamma$ is trivalent and $x = 2n$. Then $Q_I$ is equal to

$$ \cup_{\sigma \in \mathfrak{S}_{2n}} \prod_{i=1}^{2n} A_I^{(\sigma(i))}, $$

where $\mathfrak{S}_{2n}$ is the set of permutations of $2n$. We get

$$ \Delta = \sum_{\sigma \in \mathfrak{S}_{2n}} \Delta_{\sigma} $$

with

$$ \Delta_{\sigma} = \sum_{I \subseteq \underline{2n}} (-1)^{|I|} \int_{\prod_{i=1}^{2n} A_I^{(\sigma(i))}} \bigwedge_{e \in E(\Gamma)} p_e^*(\omega_I). $$

Let us compute $\Delta_{\sigma}$. For any $i \in \underline{x}$, $p_i: \check{C}(\tilde{R}_I, L; \Gamma) \to \tilde{R}_I$ denotes the projection onto the $i$th factor. When $e$ is an oriented edge from the vertex $x(e) \in V(\Gamma)$ to $y(e) \in V(\Gamma)$, we have

$$ p_e^*(\omega_I)|_{\prod_{i=1}^{2n} A_I^{(\sigma(i))}} = \sum_{\substack{j=1,\dots,g_{\sigma(x(e))} \\ \ell=1,\dots,g_{\sigma(y(e))}}} lk(z_j^{\sigma(x(e))}, z_{\ell}^{\sigma(y(e))}) p_{x(e)}^*(\eta(a_j^{\sigma(x(e))})) \wedge p_{y(e)}^*(\eta(a_{\ell}^{\sigma(y(e))})), $$

where the vertices are regarded as elements of $\underline{2n}$ via the numbering. Recall the sets $E(\Gamma)$ and $H(\Gamma)$ of edges and half-edges of $\Gamma$. For a half-edge $c$, let $v(c)$ denote the label of the vertex contained in $c$.

Let $F_{\sigma}$ denote the set of maps $f$ from $H(\Gamma)$ to $\mathbb{N}$ such that $f(c) \in \{1, 2, \dots, g_{\sigma(v(c))}\}$ for any $c \in H(\Gamma)$. For such a map $f$, $f(x(e))$ (resp. $f(y(e))$) denotes the value of $f$ at the half-edge of $e$ that contains $x(e)$ (resp. $y(e)$). We have

$$ \Delta_{\sigma} = \sum_{f \in F_{\sigma}} \left( \prod_{e \in E(\Gamma)} lk\left(z_{f(x(e))}^{\sigma(x(e))}, z_{f(y(e))}^{\sigma(y(e))}\right) \right) I(f) $$

with

$$ I(f) = \int_{\prod_{i=1}^{2n} (A^{(\sigma(i))} \cup -A^{(\sigma(i))'})} \bigwedge_{e \in E(\Gamma)} p_{x(e)}^*(\eta(a_{f(x(e))}^{\sigma(x(e))})) \wedge p_{y(e)}^*(\eta(a_{f(y(e))}^{\sigma(y(e))})). $$
We also have

$$I(f) = \prod_{i=1}^{2n} \int_{(A^{(\sigma(i))} \cup -A^{(\sigma(i))'})} \wedge \prod_{c \in H(\Gamma) : v(c)=i} p_i^*(\eta(a_{f(c)}^{\sigma(i)})),$$

where we order the exterior product's factors over the half-edges of $v^{-1}(i)$ according to the vertex-orientation of $i$. Observe

$$\int_{A^{(\sigma(i))} \cup (-A^{(\sigma(i))'})} \wedge \prod_{c \in v^{-1}(i)} \eta(a_{f(c)}^{\sigma(i)}) = \mathcal{I}_{A^{(\sigma(i))} A^{(\sigma(i))'}} \left( \bigotimes_{c \in v^{-1}(i)} a_{f(c)}^{\sigma(i)} \right),$$

where the factors of the tensor product are ordered according to the vertex-orientation of $i$, again. Indeed, the closed form $\eta(a_{f(c)}^{\sigma(i)})$ is dual to the homology class $\partial_{MV}^{-1}(a_{f(c)}^{\sigma(i)})$ in $A^{(\sigma(i))} \cup (-A^{(\sigma(i))'})$, with the notation introduced before Theorem 18.6.

Summarizing, we get

$$\Delta_{\sigma} = \sum_{f \in F_{\sigma}} \left( \left( \prod_{e \in E(\Gamma)} lk(z_{f(x(e)))^{\sigma(x(e))}, z_{f(y(e)))^{\sigma(y(e))}) \right) \left( \prod_{i \in \underline{2n}} \mathcal{I}_{A^{(\sigma(i))} A^{(\sigma(i))'}} \left( \bigotimes_{c \in v^{-1}(i)} a_{f(c)}^{\sigma(i)} \right) \right) \right).$$

We may restrict the sum to the subset $\tilde{F}_{\sigma}$ of $F_{\sigma}$ consisting of the maps $f$ of $F_{\sigma}$ that restrict to $v^{-1}(i)$ as injections for any $i$.

Finally, $\Delta$ is a sum running over all the ways of renumbering the vertices of $\Gamma$ by elements of $\underline{x}$ (via $\sigma$) and of coloring the half-edges $c$ of $v^{-1}(i)$ by three distinct curves $z_{f(c)}^{\sigma(i)}$ via $f$. In particular, a pair $(\sigma, f)$ provides a tripod

$$\begin{tikzpicture}[baseline=(current bounding box.center), scale=0.7, every node/.style={scale=0.7}] 
  \node (z_1) at (0,0) {$z_1^y$};
  \node (z_2) at (1,0) {$z_2^y$};
  \node (z_3) at (2,0) {$z_3^y$};
  \draw[->] (z_1) -- (z_2);
  \draw[->] (z_2) -- (z_3);
\end{tikzpicture}$$

such that $1 \le j < k < \ell \le g_y$ for any $y \in \underline{x}$, and it provides a pairing of the ends of the univalent vertices of the tripods, giving rise to the graph $\Gamma$ with a possibly different vertex-orientation. The vertices of the obtained graph are furthermore numbered by the numbering of the vertices of $\Gamma$, and its edges are identified with the original edges of $\Gamma$.

Fix a set of tripods associated to the elements of $\underline{x}$ as above and a pairing of their univalent vertices giving rise to $\Gamma$ as a nonoriented graph. Then there are $|\text{Aut}(\Gamma)|$ ways of numbering its vertices and edges to get a graph isomorphic to $\Gamma$. So the pairing occurs $|\text{Aut}(\Gamma)|$ times. $\square$

PROOF OF THEOREM 18.6 UP TO THE UNPROVED CLAIMS OF THIS SECTION
... RESTATED PRECISELY AT THE END OF THE PROOF: Lemma 18.34 and Proposition 7.25 easily imply

$$ \sum_{I \subseteq \underline{x}} (-1)^{|I|} Z_n(C_I, L, (\omega_I)) = 0 \quad \text{if } 2n < x, \\ = [\langle\langle \coprod_{i \in \underline{x}} T(\mathcal{I}_{A^{(i)}A^{(i)'}}) \rangle\rangle] \quad \text{if } 2n = x. $$

Theorem 19.17 (or Theorem 12.7 if pseudo-parallelizations are not required) implies

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

Set

$$ Y_n = \sum_{I \subseteq \underline{x}} (-1)^{|I|} \left( \exp \left( -\frac{1}{4} p_1(\tau_I) \beta \right) Z \right)_n (C_I, L, (\omega_I)), $$

where $(\cdot)_n$ stands for the degree $n$ part. Note that the framing corrections $\prod_{j=1}^k (\exp(-I_\theta(K_j, \tau_I)\alpha)\#_j)$ do not depend on $I$. Therefore, it suffices to prove

$$ Y_n = \sum_{I \subseteq \underline{x}} (-1)^{|I|} Z_n(C_I, L, (\omega_I)) $$

when $2n \le x$. We have

$$ \left( \exp \left( -\frac{1}{4} p_1(\tau_I) \beta \right) Z \right)_n (C_I, L, (\omega_I)) = Z_n(C_I, L, (\omega_I)) \\ + \sum_{j<n} Z_j(C_I, L, (\omega_I)) P_{n-j}(I), $$

with

$$ P_{n-j}(I) = \left( \exp\left(-\frac{1}{4}\left(p_1(\tau) + \sum_{i \in I} p(i)\right)\left(\beta_1 + \beta_3 + \dots\right)\right) \right)_{n-j}, $$

thanks to Lemma 18.32.

This element $P_{n-j}(I)$ of $\mathcal{A}_{n-j}(\emptyset)$ can be expanded as a combination $\sum m_{\Gamma,g,K}[\Gamma]$, where

* the $\Gamma$ are degree $(n-j)$ Jacobi diagrams,

* the $K$ are subsets of $I$ with cardinality at most $n-j$,

* the $m_{\Gamma,g,K}$ are monomials of degree at most $(n-j)$ in $p_1(\tau)$ and in the $p(i)$ for $i \in K$,
• the $m_{\Gamma,g,K}$ do not depend on $I$,

• for every $i \in K$, the variable $p(i)$ actually occurs in the monomial $m_{\Gamma,g,K}$, and

• for a subset $J$ of $\underline{x}$, $m_{\Gamma,g,K}[\Gamma]$ appears in $P_{n-j}(J)$ if and only if $K$ is a subset of $J$.

Thus, we can write the sum of the undesired terms in $Y_n$ by factoring out the $m_{\Gamma,g,K}[\Gamma]$. The factor of $m_{\Gamma,g,K}[\Gamma]$ is

$$ \sum_{I: K \subseteq I \subseteq \underline{x}} (-1)^{|I|} Z_j(C_I, L, (\omega_I)). $$

This sum actually runs over the subsets of $\underline{x} \setminus K$. The cardinality of $\underline{x} \setminus K$ is at least $x + j - n$. The inequalities $2n \le x$ and $j < n$ imply $j < n \le x - n$, and hence $2j < x + j - n$. Therefore, the beginning of the proof ensures that the above factor of $m_{\Gamma,g,K}[\Gamma]$ is zero. Hence we have $Y_n = \sum_{I \subseteq \underline{x}} (-1)^{|I|} Z_n(C_I, L, (\omega_I))$. This concludes the reduction of the proof of Theorem 18.6 to the proof of Proposition 18.33 given in Chapter 20, and to the proofs that pseudo-parallelizations and associated propagating forms exist and satisfy the announced properties, given in Chapter 19. $\square$

## 18.8 Mixed universality statements

We can mix the statements of Theorem 17.30 and 18.6 to get the following statement, which covers both of them.

**Theorem 18.35.** Let $y, z \in \mathbb{N}$. Recall $\underline{y} = \{1, 2, \dots, y\}$. Set $(\underline{z} + y) = \{y + 1, y + 2, \dots, y + z\}$. Let $L$ be a singular $q$-tangle representative in a rational homology cylinder $C$, whose double points are numbered by $y$ and sitting in balls $B_b$ of desingularizations for $b \in \underline{y}$. For a subset $I$ of $\underline{y}$, let $L_I$ denote the $q$-tangle obtained from $L$ by performing negative desingularizations on double points of $I$ and positive ones on double points of $\underline{y} \setminus I$ in the balls $B_b$. Let $\bigsqcup_{i=y+1}^{y+z} A^{(i)}$ be a disjoint union of rational homology handlebodies embedded in $C \setminus (L \cup_{b=1}^y B_b)$. Let $(A^{(i)}/A^{(i)})$ be rational LP-surgeries in $C$. Set $X = [C, L; (A^{(i)}/A^{(i)})_{i \in \underline{z}+\underline{y}}]$ and, using Notation 17.29,

$$ \overline{Z}_n(X) = \sum_{I \subseteq \underline{y} + \underline{z}} (-1)^{|I|} \overline{Z}_n(C((A^{(i)}/A^{(i)})_{i \in I \cap (\underline{z} + \underline{y})}), L_I \cap \underline{y}). $$
If $2n < 2y + z$, then $\overline{\mathbb{Z}}_n(X)$ vanishes. If $2n = 2y + z$, then we have

$$ \overline{\mathbb{Z}}_n(X) = \left[ \left\langle \left\langle \bigsqcup_{i \in \underline{z}+y} T(\mathcal{I}_{A^{(i)}A^{(i)'}}) \right\rangle \right\rangle_{\tilde{R}(\mathcal{C})} \sqcup \Gamma_{\mathcal{C}}(L) \right]. $$

PROOF ASSUMING THEOREM 18.6: Write $(\mathcal{C}, L)$ as a product of a tangle $L_1$ of the form

$$ \left( \left( \bigcirc_{i \in \underline{z}+y} \cdots \bigcirc_{i \in \underline{z}+y} \middle| \cdots \middle| \right) \right) $$

in the standard rational homology cylinder $D_1 \times [0, 1]$, by a nonsingular tangle $L_2$ in $\mathcal{C}$, so that $\bigsqcup_{i=y+1}^{y+z} A^{(i)}$ is in the latter factor. This can be achieved by moving the double points below. Then Proposition 17.32 and Theorem 18.6 imply that

$$ \sum_{I \subseteq \underline{y}+\underline{z}} (-1)^{|I|} \mathcal{Z}_n^f \left( \mathcal{C}\left(\left(A^{(i)'}/A^{(i)}\right)_{i \in I \cap (\underline{z}+\underline{y})}\right), L_{I \cap \underline{y}} \right) $$

satisfies the conclusions of the statement with $\mathcal{Z}^f$ instead of $\overline{\mathbb{Z}}$, by functionality. Now, to prove Theorem 18.35 it suffices to “multiply the proof of Theorem 17.30 (at the end of Section 17.6) by $[\langle\langle\bigsqcup_{i \in \underline{z}+y} T(\mathcal{I}_{A^{(i)}A^{(i)'}})\rangle\rangle_{\tilde{R}(\mathcal{C})}]$” $\square$

In order to prove more interesting mixed universality properties, we say more about the normalization of the propagating forms of Proposition 18.33.

Recall that $[-4, 4] \times \partial A^{(i)}$ denotes a regular neighborhood of $\partial A^{(i)}$ embedded in $\mathcal{C}$, that intersects $A^{(i)}$ as $[-4, 0] \times \partial A^{(i)}$. All the neighborhoods $[-4, 4] \times \partial A^{(i)}$ are disjoint from each other and from $L$. Throughout this paragraph, we use the corresponding coordinates on the image of this implicit embedding.

For $t \in [-4, 4]$, set

$$ A_t^{(i)} = \begin{cases} A^{(i)} \cup ([0, t] \times \partial A^{(i)}) & \text{if } t \ge 0 \\ A^{(i)} \setminus ([t, 0] \times \partial A^{(i)}) & \text{if } t \le 0. \end{cases} $$

We have $\partial A_t^{(i)} = \{t\} \times \partial A^{(i)}$.

For $i \in \underline{x}$, choose a basepoint $p^i$ in $\partial A^{(i)}$ outside the neighborhoods $a_j^i \times [-1, 1]$ of the $a_j^i$ and outside neighborhoods $z_j^i \times [-1, 1]$ the $z_j^i$. Fix a path $[p^i, q^i]$ from $p^i$ to a point $q^i$ of $\partial \mathcal{C}$ in

$$ \mathcal{C} \setminus \left( L \cup \mathrm{Int}(A^{(i)}) \cup \bigcup_{k: k \neq i} A_4^{(k)} \right) $$

so that the paths $[p^i, q^i]$ are pairwise disjoint. Choose a closed 2-form $\omega(p^i)$ on $(C_1(R_I) \setminus \mathrm{Int}(A^{(i)}))$ such that
• the integral of $\omega(p^i)$ along a closed surface of $(C_I \setminus \text{Int}(A^{(i)}))$ is its algebraic intersection with $[p^i, q^i]$,

• the support of $\omega(p^i)$ intersects $(C_I \setminus \text{Int}(A^{(i)}))$ inside a tubular neighborhood of $[p^i, q^i]$ disjoint from

$$ (\cup_{k:k \neq i} A_4^k) \cup L \cup \left( [0, 4] \times \left( \cup_{j=1}^{g_i} \left( (a_j^i \times [-1, 1]) \cup (z_j^i \times [-1, 1]) \right) \right) \right). $$

• $\omega(p^i)$ restricts as the usual volume form $\omega_{S^2}$ on $\partial C_1(R_I) = S^2$.

For $i \in \underline{x}$, for $j = 1, \dots, g_i$, the curve $\{4\} \times a_j^i$ bounds a rational chain $\Sigma(a_j^i)$ in $A_4^{(i)}$ and a rational chain $\Sigma'(a_j^i)$ in $A_4^{(i)'}$. When viewed as a chain in $C_I$, such a chain is denoted by $\Sigma_I(a_j^i)$. We have $\Sigma_I(a_j^i) = \Sigma(a_j^i)$ if $i \notin I$, and $\Sigma_I(a_j^i) = \Sigma'(a_j^i)$ if $i \in I$. The form $\eta(a_j^i)$ supported on $[-4, 4] \times a_j^i \times [-1, 1]$ in $A_{I,4}^{(i)} \setminus A_{I,-4}^{(i)}$ may be expressed as $\eta(a_j^i) = p_{[-1,1]}^*(\eta_{[-1,1]})$ there. Thus, it extends naturally to $A_{I,4}^{(i)} = (A_I^{(i)})_4$, as a closed form dual to the chain $\Sigma_I(a_j^i)$.

For $i \in \underline{x}$, for $j = 1, \dots, g_i$, $z_j^i$ bounds a rational chain in $C_I$. Therefore, it cobounds a rational chain $\Sigma_I(\tilde{z}_j^i)$ in $(C_I \setminus \mathring{A}_I^{(i)}) \setminus (\cup_{i=1}^x [p^i, q^i])$ with a combination of $a_\ell^i$ with rational coefficients. We have

$$ \partial\Sigma_I(\tilde{z}_j^i) = z_j^i - \sum_{j=1}^{g_i} lk_e(z_j^i, \{-1\} \times z_\ell^i) a_\ell^i = \tilde{z}_j^i. $$

Furthermore, $\Sigma_I(\tilde{z}_j^i)$ may be assumed to intersect $\mathring{A}_I^{(k)}$ as

$$ \sum_{m=1}^{g_k} lk(z_j^i, z_m^k) \Sigma_I(a_m^k), $$

for $k \neq i$. There is a closed one-form $\eta_I(z_j^i)$ dual to $\Sigma_I(\tilde{z}_j^i)$ in $(C_I \setminus \mathring{A}_I^{(i)})$, such that $\eta_I(z_j^i)$ is supported near $\Sigma_I(\tilde{z}_j^i)$ and outside the supports of $\omega(p^i)$ and of the other $\omega(p^k)$, and we have

$$ \eta_I(z_j^i) = \sum_{m=1}^{g_k} lk(z_j^i, z_m^k) \eta(a_m^k) $$

on $\mathring{A}_I^{(k)}$ for $k \neq i$. The integral of $\eta_I(z_j^i)$ along a closed curve of $(C_I \setminus \mathring{A}_I^{(i)})$ is its linking number with $z_j^i$ in $C_I$.

We will prove the following proposition in Chapter 20.
**Proposition 18.36.** The antisymmetric propagating forms $\omega_I$ of $(C_2(R_I), \tau_I)$ of Proposition 18.33 can be chosen so that

1. for every $i \in \underline{x}$, the restriction of $\omega_I$ to

$$ \left( A_I^{(i)} \times \left( C_1(R_I) \setminus A_{I,3}^{(i)} \right) \right) \subset C_2(R_I) $$

is equal to

$$ \sum_{j \in g_i} p_1^*(\eta_I(a_j^i)) \wedge p_2^*(\eta_I(z_j^i)) + p_2^*(\omega(p^i)), $$

where $p_1$ and $p_2$ respectively denote the first and second projection of $A_I^{(i)} \times (C_1(R_I) \setminus A_{I,3}^{(i)})$ to $C_1(R_I)$, and

2. for every $i$, for any $j \in \{1, 2, \dots, g_i\}$, we have

$$ \int_{\Sigma_I(a_j^i) \times p^i} \omega_I = 0, $$

where $p^i \in \partial A_I^{(i)}$ and $\partial \Sigma_I(a_j^i) \subset \{4\} \times \partial A_I^{(i)}$.

A two-leg Jacobi diagram is a uni-trivalent Jacobi diagram with two univalent vertices, called *legs*. When these legs are colored by possibly noncompact connected components $K_j$ of a tangle $L$, a two-leg diagram gives rise to a diagram on the domain $\mathcal{L}$ of the tangle by attaching the legs to the corresponding components. The class in $\mathcal{A}(\mathcal{L})$ of this diagram is well-defined. Indeed, according to Lemma 6.26, Jacobi diagrams with one univalent vertex vanish in $\mathcal{A}(\mathcal{L})$. Therefore, the STU relation guarantees that if the same noncompact component colors the two legs, changing their order with respect to the orientation component does not change the diagram class. (See also Lemma 12.26.)

Generalize the contraction of trivalent graphs associated to LP-surgeries of Notation 18.5 to graphs with legs as follows.

**Notation 18.37.** Let $L: \mathcal{L} \hookrightarrow \mathcal{C}$ be a long tangle representative in a rational homology cylinder $\mathcal{C}$.

Let $G$ be a graph with oriented trivalent vertices and with two kinds of univalent vertices, the *decorated ones* and the *legs*, where the components of the legs of $G$ are edges from a leg to a decorated univalent vertex. The *legs* are univalent vertices on $\mathcal{L}$. The decorated univalent vertex in a leg segment is decorated with the leg component. The other *decorated univalent vertices*
of $G$ are decorated with disjoint curves of $\tilde{R} = \tilde{R}(\mathcal{C})$ disjoint from the image of $L$. Such a curve $c$ bounds a compact oriented surface $\Sigma(c)$ in $\mathcal{C}$, and its linking number with a component $K_u$ of $L$ is the algebraic intersection $\langle K_u, \Sigma(c) \rangle$. Let $\tilde{P}(G)$ be the set of partitions of the set of decorated univalent vertices of $G$ in disjoint pairs.

For $p \in \tilde{P}(G)$, identifying the two decorated vertices of each pair provides a vertex-oriented Jacobi diagram $\Gamma_p$ on $\mathcal{L}$. Multiplying it by the product $\ell(p)$ over the pairs of $p$ of the linking numbers of the curves that decorate the two vertices yields an element $[\ell(p)\Gamma_p]$ of $\mathcal{A}(\mathcal{L})$.

Define

$$ \langle\langle G \rangle\rangle = \tilde{p} \left( \sum_{p \in \tilde{P}(G)} [\ell(p) \Gamma_p] \right) \in \check{\mathcal{A}}(\mathcal{L}) $$

Extend this contraction to linear combinations of graphs, linearly. Assume that the components of $L$ are numbered in $\underline{k}$. To a pair $(u, v)$ of $\underline{k}$, associate the univalent graph $G_{uv} = \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}] \draw[->] (0,0) -- (1,0); \draw[->] (0,0) -- (0.5,0.866); \draw[->] (0,0) -- (0.5,-0.866); \node[left] at (-0.5,0) {$v$}; \node[right] at (1.5,0) {$u$}; \node[above] at (0.27,0.5) {$G_{uv}$}; \end{tikzpicture}$ consisting of two distinct segments, the first one with its leg on $K_u$, and the second one with its leg on $K_v$. The legs are considered as distinct even if there is an automorphism of $G_{uu} = \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}] \draw[->] (0,0) -- (1,0); \draw[->] (0,0) -- (0.5,0.866); \draw[->] (0,0) -- (0.5,-0.866); \node[left] at (-0.5,0) {$v$}; \node[right] at (1.5,0) {$u$}; \node[above] at (0.27,0.5) {$G_{uu}$}; \end{tikzpicture}$ that maps a leg to the other when $K_u$ is a circle. We think of the leg of the first segment as the first leg of $G_{uv}$, and the other leg is the second leg of $G_{uv}$.

For a finite collection $(A^{(i)\prime}/A^{(i)})_{i \in \underline{2x}}$ of disjoint rational LP-surgeries in $\check{R}(\mathcal{C}) \setminus L$, define the element

$$ [\Gamma_{(2)}(\mathcal{C}, L; (A^{(i)\prime}/A^{(i)})_{i \in \underline{2x}})] = \frac{1}{2} \sum_{(u,v) \in \underline{k}^2} \left[ \langle\langle \bigsqcup_{i \in \underline{2x}} T(\mathcal{I}_{A^{(i)}A^{(i)'}}) \sqcup G_{uv} \rangle\rangle \right] $$

of $\check{\mathcal{A}}(\mathcal{L})$.

**Examples 18.38.** Assume

$$ T(\mathcal{I}_{A^{(1)}A^{(1)'}}) = \left\langle \begin{array}{c} z_3^1 \\ z_2^1 \\ z_1^1 \end{array} \right\rangle \quad \text{and} \quad T(\mathcal{I}_{A^{(2)}A^{(2)'}}) = \left\langle \begin{array}{c} z_3^2 \\ z_2^2 \\ z_1^2 \end{array} \right\rangle, $$

where $lk(z_i^1, z_j^2) = 0$ as soon as $i \neq j$, and $lk(z_i^1, K_u) = lk(z_i^2, K_u) = 0$ if $i \neq 3$. If $K_u$ is an interval, then we get

$$ \langle\langle \bigsqcup_{i \in \underline{2}} T(\mathcal{I}_{A^{(i)}A^{(i)'}}) \sqcup G_{uu} \rangle\rangle = -2lk(z_1^1, z_1^2)lk(z_2^1, z_2^2)lk(z_3^1, K_u)lk(z_3^2, K_u) \left[ \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}] \draw[->] (0,0) -- (1,0); \draw[->] (0,0) -- (0.5,0.866); \draw[->] (0,0) -- (0.5,-0.866); \node[left] at (-0.5,0) {$v$}; \node[right] at (1.5,0) {$u$}; \node[above] at (0.27,0.5) {$G_{uu}$}; \end{tikzpicture} \right]. $$

If $K_u$ is a circle, then we get

$$ \langle\langle \bigsqcup_{i \in \underline{2}} T(\mathcal{I}_{A^{(i)}A^{(i)'}}) \sqcup G_{uu} \rangle\rangle = -2lk(z_1^1, z_1^2)lk(z_2^1, z_2^2)lk(z_3^1, K_u)lk(z_3^2, K_u) \left[ \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}] \draw[->] (0,0) -- (1,0); \draw[->] (0,0) -- (0.5,0.866); \draw[->] (0,0) -- (0.5,-0.866); \node[left] at (-0.5,0) {$v$}; \node[right] at (1.5,0) {$u$}; \node[above] at (0.27,0.5) {$G_{uu}$}; \end{tikzpicture} \right]. $$
**Theorem 18.39.** Let $L: \mathcal{L} \hookrightarrow \mathcal{C}$ be a long tangle representative in a rational homology cylinder $\mathcal{C}$. Let $\bigsqcup_{i=1}^{2x} A^{(i)}$ be a disjoint union of rational homology handlebodies embedded in $\mathcal{C} \setminus L$. Let $(A^{(i)\prime}/A^{(i)})$ be rational LP-surgeries in $\mathcal{C}$. Set $X = [\mathcal{C}, L; (A^{(i)\prime}/A^{(i)})_{i \in \underline{2x}}]$ and

$$ \check{Z}_n(X) = \sum_{I \subseteq \underline{2x}} (-1)^{|I|} \check{Z}_n(C_I, L), $$

where $C_I = C((A^{(i)\prime}/A^{(i)})_{i \in I})$ is the rational homology cylinder obtained from $\mathcal{C}$ by performing the LP-surgeries that replace $A^{(i)}$ by $A^{(i)\prime}$ for $i \in I$. Assume that $x \neq 0$. If $n < x + 1$, then $\check{Z}_n(X)$ vanishes. If $n = x + 1$, then we have

$$ \check{Z}_n(X) = [\Gamma_{(2)}(\mathcal{C}, L; (A^{(i)\prime}/A^{(i)})_{i \in \underline{2x}})], $$

where $\check{Z}$ is defined in Proposition 17.28. See also Notation 6.19 and Notation 18.37.

**Remark 18.40.** This result holds modulo 1T when $x = 0$.

**PROOF OF THEOREM 18.39 ASSUMING PROPOSITION 18.36:** Recall that $\check{Z}$ takes its values in $\check{\mathcal{A}}(\mathcal{L})$, where the diagrams with connected trivalent components vanish. Therefore, Theorem 18.6 implies the result when $n < x + 1$. Assume that $n = x + 1$. Since the framing correction terms involving $\beta$ vanish in $\check{\mathcal{A}}(\mathcal{L})$, and since the other correction terms are the same for all the $\check{Z}(C_I, L)$, we get

$$ \check{Z}_{x+1}(X) = \sum_{I \subseteq \underline{2x}} (-1)^{|I|} \check{Z}_n(C_I, L, (\omega_I)). $$

Let $\Gamma$ be a Jacobi diagram of degree $x+1$ on $\mathcal{L}$ that contributes to $\check{Z}_{x+1}(X)$. Since its class does not vanish in $\check{\mathcal{A}}_{x+1}(\mathcal{L})$, each component of $\Gamma$ must contain at least two univalent vertices because one-leg diagrams vanish in $\check{\mathcal{A}}(\mathcal{L})$. Furthermore, as in the proof of Lemma 18.34, the configurations must involve at least one point in some $A_I^{(i)}$, for each $i$ in $\underline{2x}$. Such a point must be a trivalent vertex position. Therefore, the graph $\Gamma$ has at least $2x$ trivalent vertices. Finally, $\Gamma$ must be a connected two-leg Jacobi diagram of degree $x+1$ on $\mathcal{L}$. Assume that the univalent vertices of $\Gamma$ are on components $K_u$ and $K_v$. Order the set of univalent vertices of $\Gamma$ so that the first is on $K_u$ and the second is on $K_v$. Let $\Gamma^U$ denote the graph $\Gamma$ equipped with this order. Number the trivalent vertices of $\Gamma$ in $\underline{2x}$. Orient the open configuration space $\check{C}(\check{R}_I, L; \Gamma)$ as an open oriented submanifold of $K_u \times K_v \times \check{R}_I^{2x}$ with respect to the order of $V(\Gamma)$ induced by the numbering of its elements. Orient the edges
and the trivalent vertices of Γ so that the induced orientation of $\tilde{C}(\tilde{R}_I, L; \Gamma)$ matches the previous one.

As in the proof of Lemma 18.34, we have

$$ \Delta(\Gamma) = \sum_{I \subseteq \underline{2x}} (-1)^{|I|} I(R_I, L, \Gamma, o(\Gamma), (\omega_I)) [\Gamma] = \sum_{\sigma \in \mathfrak{S}_{2x}} \Delta_{\sigma} [\Gamma], $$

with

$$ \Delta_{\sigma} = \sum_{I \subseteq \underline{2x}} (-1)^{|I|} \int_{D(I, \sigma)} \bigwedge_{e \in E(\Gamma)} p_e^*(\omega_I), $$

and

$$ D(I, \sigma) = K_u \times K_v \times \prod_{i=1}^{2x} A_I^{(\sigma(i))}, $$

when $K_u \neq K_v$, or when $K_u$ is a closed component. When $K_u = K_v$ and $K_u$ is a long component, the factor $K_u \times K_v$ must be replaced by one of its two subsets of pairs with a fixed order on $K_u$. When $e$ is an oriented edge between two trivalent vertices, recall the expression of $p_e^*(\omega_I)$ from Proposition 18.33. Without loss of generality, assume that the legs are the first half-edges of their edges. With the projections $p_i : \tilde{C}(\tilde{R}_I, L; \Gamma) \to \tilde{R}_I$, Proposition 18.36 implies

$$ p_e^*(\omega_I)|_{D(I, \sigma)} = \sum_{j \in \underline{g_{\sigma(y(e))}}} p_{x(e)}^* \left( \eta(z_j^{\sigma(y(e))}) \right) \wedge p_{y(e)}^* \left( \eta(a_j^{\sigma(y(e))}) \right) $$

for an edge from a leg $x(e)$ to a trivalent vertex $y(e)$. In this case, the edge $e$ is associated to $x(e)$ and denoted by $e(x(e))$. Let $H_T(\Gamma)$ denote the subset of $H(\Gamma)$ consisting of the half-edges that contain a trivalent vertex, and let $E_T(\Gamma)$ denote the set of edges of $\Gamma$ between trivalent vertices. Let $F_{\sigma}$ denote the set of maps $f$ from $H_T(\Gamma)$ to $\mathbb{N}$ such that $f(c)$ belongs to $\{1, 2, \dots, g_{\sigma(v(c))}\}$ for any $c \in H_T(\Gamma)$. We have

$$ \Delta_{\sigma} = \sum_{f \in F_{\sigma}} \left( \prod_{e \in E_T(\Gamma)} lk \left( z_{f(x(e))}^{\sigma(x(e))}, z_{f(y(e))}^{\sigma(y(e))} \right) \right) I(f), $$

where $I(f)$ is equal to

$$ \int_{K_u} \eta(z_{f(y(u))}^{\sigma(y(u))}) \int_{K_v} \eta(z_{f(y(v))}^{\sigma(y(v))}) \times \prod_{i=1}^{2x} \int_{(A^{(\sigma(i))} \cup -A^{(\sigma(i))})'} \bigwedge_{c \in H(\Gamma): v(c)=i} p_i^* (\eta(a_{f(c)}^{\sigma(i)})), $$
when $K_u \neq K_v$, or when $K_u$ is a closed component. Recall

$$ \int_{K_u} \eta \left( z_{f(y(u))}^{\sigma(y(u))} \right) = lk \left( K_u, z_{f(y(u))}^{\sigma(y(u))} \right). $$

Summarizing, when $K_u \neq K_v$ or when $K_u$ is a closed component, we have

$$ \Delta_{\sigma} = \sum_{f \in F_{\sigma}} \left( \left( \prod_{e \in E(\Gamma)} lk(e; f) \right) \left( \prod_{i \in \underline{2x}} \mathcal{I}_{A^{(\sigma(i))} A^{(\sigma(i))'}} \left( \bigotimes_{c \in v^{-1}(i)} a_{f(c)}^{\sigma(i)} \right) \right) \right), $$

where $lk(e; f) = lk(z_{f(x(e))}^{\sigma(x(e))}, z_{f(y(e))}^{\sigma(y(e))})$ when $e \in E_T(\Gamma)$, and

$$ lk(e; f) = lk \left( K_{x(e)}, z_{f(y(e))}^{\sigma(y(e))} \right) $$

when $x(e)$ is univalent.

Finally, $\Delta(\Gamma)$ is a sum, running over all the ways of renumbering the trivalent vertices of $\Gamma$ by elements of $\underline{2x}$ (via $\sigma$), and of coloring the half-edges $c$ of $v^{-1}(i)$ by three distinct curves $z_{f(c)}^{\sigma(i)}$ via $f$.

In particular, a pair $(\sigma, f)$ provides a tripod

<figure><img src="image_150.png" /></figure>

for any $y \in \underline{2x}$ such that $1 \le j < k < \ell \le g_y$. It also provides a pairing of the ends of the univalent vertices of the tripods and of the legs on $K_u$ and $K_v$ (the first one and the second one when $v = u$), which gives rise to the graph $\Gamma$ with a possibly different vertex-orientation. The vertices of the obtained graph are furthermore numbered by the numbering of the vertices of $\Gamma$, and its edges are identified with the original edges of $\Gamma$. The order of $U(\Gamma)$ is induced by the order on the legs of $G_{uv}$.

Let $\mathrm{Aut}(\Gamma^U)$ denote the set of automorphisms of $\Gamma$ that fix the univalent vertices of $\Gamma$ (pointwise). (The set $\mathrm{Aut}(\Gamma^U)$ is distinct from $\mathrm{Aut}(\Gamma)$ if and only if $u=v$, $K_u$ is a closed component, and there exists an automorphism of $\Gamma$ that exchanges its two univalent vertices.)

Fix a set of tripods associated to the elements of $\underline{2x}$ as above and a pairing of their univalent vertices and the legs on $K_u$ and $K_v$ (which are distinguished as the first one and the second one when $v = u$). Then there are exactly $|\mathrm{Aut}(\Gamma^U)|$ ways of numbering its vertices and edges to get a graph isomorphic to $\Gamma^U$ by an isomorphism that fixes the univalent vertices. So the pairing occurs $|\mathrm{Aut}(\Gamma^U)|$ times.
Let $G$ be a family of $2x$ tripods

$$ \leftarrow z_{\ell}^{y} \quad z_{k}^{y} \quad z_{j}^{y}, $$

one for each $y \in \underline{2x}$, such that $1 \le j < k < \ell \le g_y$. Define

$$ [\langle\langle G \sqcup G_{uv} \rangle\rangle_{\Gamma^U}] = \sum_{p \in \tilde{P}(G \sqcup G_{uv}): \Gamma_p \text{ isomorphic to } \Gamma^U} [\ell(p)\Gamma_p], $$

where the sum runs over the $p$ such that $\Gamma_p$ is isomorphic to $\Gamma^U$, as a nonoriented uni-trivalent graph on $\mathcal{L}$, equipped with a fixed order on $U(\Gamma)$, with the notation introduced before the statement of Theorem 18.39. Similarly define

$$ [\langle\langle G \sqcup G_{uv} \rangle\rangle_{\Gamma}] = \sum_{p \in \tilde{P}(G \sqcup G_{uv}): \Gamma_p \text{ isomorphic to } \Gamma} [\ell(p)\Gamma_p], $$

where the sum runs over the $p$ such that $\Gamma_p$ is isomorphic to $\Gamma$, as a nonoriented uni-trivalent graph on $\mathcal{L}$. (We forget the order on $U(\Gamma)$.)

We get

$$ \Delta(\Gamma) = |\mathrm{Aut}(\Gamma^U)| \left[ \left\langle\left\langle \bigsqcup_{i \in \underline{2x}} T(\mathcal{I}_{A^{(i)}A^{(i)'}}) \sqcup G_{uv} \right\rangle\right\rangle_{\Gamma^U} \right]. $$

If $u \ne v$, then $\Gamma^U$ and $\Gamma$ coincide, and we have

$$ \frac{\Delta(\Gamma)}{|\mathrm{Aut}(\Gamma)|} = \frac{1}{2} \left( \left[ \left\langle\left\langle \bigsqcup_{i \in \underline{2x}} T(\mathcal{I}_{A^{(i)}A^{(i)'}}) \sqcup G_{uv} \right\rangle\right\rangle_{\Gamma} \right] + \left[ \left\langle\left\langle \bigsqcup_{i \in \underline{2x}} T(\mathcal{I}_{A^{(i)}A^{(i)'}}) \sqcup G_{vu} \right\rangle\right\rangle_{\Gamma} \right] \right). $$

Assume that $u = v$ and that $K_u$ is a circle. If there exists an automorphism of $\Gamma$ that exchanges its univalent vertices, then we have $|\mathrm{Aut}(\Gamma)| = 2|\mathrm{Aut}(\Gamma^U)|$ and $\langle\langle\cdot\rangle\rangle_{\Gamma^U} = \langle\langle\cdot\rangle\rangle_{\Gamma}$. Otherwise, we have $|\mathrm{Aut}(\Gamma)| = |\mathrm{Aut}(\Gamma^U)|$ and $[\langle\langle\cdot\rangle\rangle_{\Gamma}] = 2[\langle\langle\cdot\rangle\rangle_{\Gamma^U}]$. So we obtain

$$ \Delta(\Gamma) = \frac{1}{2} |\mathrm{Aut}(\Gamma)| \left[ \left\langle\left\langle \bigsqcup_{i \in \underline{2x}} T(\mathcal{I}_{A^{(i)}A^{(i)'}}) \sqcup G_{uu} \right\rangle\right\rangle_{\Gamma} \right] $$

in any case.

It remains to study the case in which the univalent vertices of $\Gamma$ belong to the same noncompact component $K_u$. In this case, we compute the sum
$\Delta(\Gamma) + \Delta(\Gamma^s)$, where $\Gamma^s$ is obtained from $\Gamma$ by exchanging the order of its univalent vertices on $K_u$.

Again, we find

$$ \Delta(\Gamma) + \Delta(\Gamma^s) = |\mathrm{Aut}(\Gamma)| \left[ \left\langle \left\langle \bigsqcup_{i \in \mathcal{I}} T(\mathcal{I}_{A^{(i)}A^{(i)'}}) \sqcup G_{uu} \right\rangle \right\rangle_{\Gamma} \right], $$

where the contraction $\langle\langle\cdot\rangle\rangle_{\Gamma}$ keeps only the graphs that are isomorphic to $\Gamma$ (as a graph with an ordered pair of free legs). (Recall $[\Gamma^s] = [\Gamma]$ in $\mathcal{A}(\mathcal{L})$). If $\Gamma$ and $\Gamma^s$ are isomorphic, then we have $\Delta(\Gamma) = \Delta(\Gamma^s)$ and

$$ \Delta(\Gamma) = \frac{1}{2} |\mathrm{Aut}(\Gamma)| \left[ \left\langle \left\langle \bigsqcup_{i \in \mathcal{I}} T(\mathcal{I}_{A^{(i)}A^{(i)'}}) \sqcup G_{uu} \right\rangle \right\rangle_{\Gamma} \right]. $$

Otherwise, we have $|\mathrm{Aut}(\Gamma)| = |\mathrm{Aut}(\Gamma^s)|$ and $[\langle\langle\cdot\sqcup G_{uu}\rangle\rangle_{\Gamma}] = [\langle\langle\cdot\sqcup G_{uu}\rangle\rangle_{\Gamma^s}]$. So we get

$$ \sum_{\substack{\Gamma \text{ as above} \\ \text{with 2 univalent vertices on } K_u}} \frac{\Delta(\Gamma)}{|\mathrm{Aut}(\Gamma)|} = \frac{1}{2} \left[ \left\langle \left\langle \bigsqcup_{i \in \mathcal{I}} T(\mathcal{I}_{A^{(i)}A^{(i)'}}) \sqcup G_{uu} \right\rangle \right\rangle_{\Gamma} \right] $$

in any case.

□

As in Section 18.4, we can associate an alternate sum of tangles to a framed embedding of a Jacobi diagram on a tangle $L$ into a rational homology cylinder.

Let $\Gamma$ be such a Jacobi diagram, whose trivalent vertices are numbered in $\underline{x}$. Let $\Sigma(\Gamma)$ be an oriented surface containing $\Gamma$ in its interior such that $\Sigma(\Gamma)$ is a regular neighborhood of $\Gamma$ in $\Sigma(\Gamma)$. Equip $\Gamma$ with its vertex-orientation induced by the orientation of $\Sigma(\Gamma)$. Embed $\Sigma(\Gamma)$ in $\mathcal{C}$, so that $L$ intersects $\Sigma(\Gamma)$ near univalent vertices as in

<figure><img src="image_151.png" /></figure>

$L$ is tangent to $\Sigma(\Gamma)$ at univalent vertices, and $L$ does not meet $\Sigma(\Gamma)$ outside such neighborhoods of the univalent vertices. Note that the embedding of $\Sigma(\Gamma)$ induces a local orientation of $L$ as in Definition 6.16. Replace a chord between two univalent vertices by a crossing change so that

<figure><img src="image_152.png" /></figure>
associated to the positive crossing change from

$$ \text{to } \text{underbrace{\Large\arrowvert}}. $$

Replace neighborhoods

$$ \text{of the edges} \text{ under} \text{ the } \text{underbrace{\Large\arrowvert}}. $$

between a trivalent vertex and a univalent vertex by neighborhoods

$$ \text{of } \text{under} \text{ the } \text{underbrace{\Large\arrowvert}}. $$

Finally, replace neighborhoods of the edges between trivalent vertices as in Section 18.4.

Thus, $\Sigma(\Gamma)$ transforms $L$ into a singular tangle $L^s$ whose double points are associated to the chords of $\Gamma$ as above, equipped with a collection of disjoint oriented surfaces

$$ \Sigma(Y) = \text{\Large\arrowvert}, $$

associated to trivalent vertices of $\Gamma$. The oriented surfaces $\Sigma(Y)$ are next thickened to become framed genus 3 handlebodies.

Define $\psi(\Sigma(\Gamma))$ to be $[C, L^s; (A^{(i)\prime}/A^{(i)})_{i \in \underline{x}}]$, where the surgeries $(A^{(i)\prime}/A^{(i)})$ associated to the trivalent vertices of $\Gamma$ are defined as in Section 18.4. Set $\tilde{Z}(\psi(\Sigma(\Gamma))) = \sum_{I \subseteq \underline{x}} (-1)^{x+|I|} \tilde{Z}_n(C_I, L^s)$, where $C_I = C((A^{(i)\prime}/A^{(i)})_{i \in I})$.

**Corollary 18.41.** Let $n$ be a positive integer. Let $\Gamma$ be a degree $n$ Jacobi diagram with two univalent vertices. Let $\Sigma(\Gamma)$ be a regular neighborhood of $\Gamma$ embedded in $C$ as above. Then we have $\tilde{Z}_{\le n}(\psi(\Sigma(\Gamma))) = [\Gamma]$.

**Proof:** This follows from Theorem 18.39 as in the proof of Lemma 18.24. $\square$

<figure><img src="image_153.png" /></figure>

**Remark 18.42.** This corollary could be true for more general Jacobi diagrams.

To finish this section, we apply Theorem 18.39 with the LP-surgery of Subsection 18.3.1 to compute the degree 2 part of $\tilde{Z}$ for a null-homologous knot and prove Theorem 18.43 below. Recall the definition of $a_2(K)$ from the statement of Proposition 18.12 and the lines that follow.

**Theorem 18.43.** Let $K$ be a null-homologous knot in a rational homology sphere $R$. Then we have

$$ \tilde{Z}_2(R, K) = \left( \frac{1}{24} - a_2(K) \right) \left[ \text{\Large\arrowvert} \right]. $$
PROOF: As in the proof of Proposition 18.12 in Subsection 18.3.2, let $K$ bound a Seifert surface $\Sigma$ equipped with a symplectic basis $(x_i, y_i)_{i \in g}$ in $R$. Let $\Sigma \times [-1, 2]$ be a collar of $\Sigma$ in $R$, and let $(A'/A) = (A'_F/A_F)$ and $(B'/B) = (B'_F/B_F)$ be the LP-surgeries of Subsection 18.3.1. Let $U$ be a meridian of $K$ passing through $d \times [-1, 2]$. According to Proposition 18.16, $(R(A'/A), U)$ is diffeomorphic to $(R, K)$. Similarly, $(R(B'/B), U)$ is diffeomorphic to $(R, -K)$, while $(R(A'/A, B'/B), U)$ is diffeomorphic to $(R, U)$. So we have

$$ \begin{aligned} \check{Z}_2([R, U; A'/A, B'/B]) &= 2\check{Z}_2(R, U) - \check{Z}_2(R, K) - \check{Z}_2(R, -K) \\ &= [\Gamma_{(2)}(R, U; A'/A, B'/B)], \end{aligned} $$

with

$$ [\Gamma_{(2)}(R, U; A'/A, B'/B)] = \frac{1}{2} [\langle\langle T(\mathcal{I}_{AA'}) \sqcup T(\mathcal{I}_{BB'}) \sqcup \text{輪} \rangle\rangle] \in \mathcal{A}_2(\mathcal{L}), $$

$$ T(\mathcal{I}_{AA'}) = \sum_{i=1}^{g} \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}] \node[draw, circle, inner sep=1.2pt] at (0,0) {};
    \node[above=0.3ex] at (0,0) {$x_i$};
    \node[below=0.3ex] at (0,0) {$y_i$};
    \node[left=0.5ex] at (-0.3,0) {$c$};
    \node[right=0.5ex] at (0.3,0) {$c$};
    \end{tikzpicture}, $$

and

$$ T(\mathcal{I}_{BB'}) = \sum_{j=1}^{g} \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}] \node[draw, circle, inner sep=1.2pt] at (0,0) {};
    \node[above=0.3ex] at (0,0) {$x_j^+$};
    \node[below=0.3ex] at (0,0) {$y_j^+$};
    \node[left=0.5ex] at (-0.3,0) {$c^+$};
    \node[right=0.5ex] at (0.3,0) {$c^+$};
    \end{tikzpicture}, $$

according to Lemma 18.18. We get

$$ [\Gamma_{(2)}(R, U; A'/A, B'/B)] = a_2(K) [\text{輪}]. $$

Since $\check{A}_2(S^1)$ is generated by the chord diagrams $\text{輪}$ and $\text{对角线}$, which are symmetric with respect to the orientation change on $S^1$, we have $\check{Z}_2(R, K) = \check{Z}_2(R, -K)$. So we obtain

$$ 2\check{Z}_2(R, U) - 2\check{Z}_2(R, K) = a_2(K) [\text{对角线}] = 2a_2(K) [\text{轮}]. $$

According to Example 7.22 and the multiplicativity of $\mathcal{Z}$ under connected sum of Theorem 10.26, we get $\check{Z}_2(R, U) = \frac{1}{24} [\text{轮}]$. $\square$

**Remark 18.44.** This theorem generalizes a result of Enore Guadagnini, Maurizio Martellini, and Mihail Mintchev in [GMM90] for the case $R = S^3$, to any rational homology sphere $R$. In the case of $S^3$, the known proof relies on the facts that $\check{Z}_2$ is of degree 2 and that the space of real-valued knot
invariants of degree at most 2 is generated by $a_2$ and a constant nonzero invariant. This uses the fact that any knot can be unknotted by crossing changes. This is no longer true in general rational homology spheres since crossing changes do not change the homotopy class. Our proof is more direct, and our result holds for any null-homologous knot in a rational homology sphere.

Recall the Conway weight system $w_C$ from Example 6.11, the Alexander polynomial of Definition 18.13, and Proposition 6.21. A long knot $\tilde{K}$ of a $\mathbb{Q}$-sphere $R(C)$ is an embedding of $\mathbb{R}$ into $\tilde{R}(C)$, whose image intersects the complement of $C$, as the vertical embedding ($j_{\mathbb{R}}: t \mapsto (0,0,t)$) does. Replace $j_{\mathbb{R}}(\mathbb{R} \setminus ]0,1[)$ by an arc of $\tilde{R} \setminus \tilde{C}$ from $(0,0,1)$ to $(0,0,0)$, which cobounds an embedded topological disk in $\tilde{R} \setminus \tilde{C}$ with an arc of $\partial C$ with the same ends. This provides a knot $\tilde{K}$, whose isotopy class is well-defined. In [Let22], David Leturcq proves the following theorem. Together with the properties of the functor $\mathcal{Z}^f$ of the third part of the book, this Leturcq theorem implies Theorem 18.43.

**Theorem 18.45 (Leturcq).** For any long knot $\tilde{K}$ in a rational homology sphere $\tilde{R}$, such that $\tilde{K}$ is null-homologous, we have the following equality in $\mathbb{R}[[h]]$:

$$ \sum_{n \in \mathbb{N}} w_C(\tilde{Z}_n(R, \tilde{K})) h^n = \Delta_{\tilde{K}}(\exp(h)). $$

Leturcq's proof of this theorem relies on a direct computation with appropriate propagating forms. In [Let23], David Leturcq obtains a similar theorem for the Bott–Cattaneo–Rossi invariants of higher dimensional knots [CR05, Let21]. This Leturcq theorem in higher dimensions generalizes a theorem of Tadayuki Watanabe [Wat07], who proved it for ribbon knots.
# Chapter 19

## More flexible definitions of $\mathcal{Z}$ using pseudo-parallelizations

This chapter presents the *pseudo-parallelizations* introduced in Section 18.7 and defined in Section 19.2. Our proof of the universality theorem 18.6 uses these generalizations of parallelizations first introduced in [Les04b, Section 4.3 and 4.2] and furthermore studied in [Les10, Section 10] and [Les13, Sections 7 to 10].

Pseudo-parallelizations allow us to give more flexible definitions for our invariants $\mathcal{Z}$ and $\tilde{\mathcal{Z}}$. In Theorem 19.17, we generalize the definition of the invariant $\mathcal{Z}$ of Theorem 12.7, and thus the definition of the $q$-tangle invariant $\mathcal{Z}^f$ of Definition 13.10, by allowing propagating forms associated with pseudo-parallelizations. We conclude the proof of Theorem 19.17 in Section 19.5. This proof involves all the previous sections.

In Chapter 21, we present variants of the definition of $\mathcal{Z}^f$ involving nonhomogeneous propagating forms or propagating chains associated with pseudo-parallelizations. We will not use these variants in the proof of Theorem 18.6.

### 19.1 Why we need pseudo-parallelizations

This section explains why a parallelization of the exterior of a $Q$-handlebody $A$ does not necessarily extend to $A'$ after a rational LP-surgery $(A'/A)$. It justifies why I could not avoid this chapter and some of its difficulties.

**Notation 19.1.** Let $\alpha$ be a smooth map from $[-1, 1]$ to $[0, 2\pi]$ such that $\alpha$ maps $[-1, -1 + \varepsilon]$ to $0$ for some $\varepsilon \in ]0, \frac{1}{8}[$, it increases from $0$ to $2\pi$ on $[-1 + \varepsilon, 1 - \varepsilon]$, and $\alpha(-u) + \alpha(u) = 2\pi$ for any $u \in [-1, 1]$.

Let $\Sigma$ be a surface, orientable or not. Let $\gamma$ be a two-sided curve properly embedded in $\Sigma$ and equipped with a collar $\gamma \times [-1, 1]$ in $\Sigma$. Let $\rho_{\alpha(u)} =$
$\rho(\alpha(u), (0, 0, 1))$ denote the rotation of $\mathbb{R}^3$ of angle $\alpha(u)$ whose axis is directed by $(0, 0, 1)$. Then $\mathcal{R}_\gamma$ denotes the map from $\Sigma$ to $SO(3)$ sending $\Sigma \setminus (\gamma \times [-1, 1])$ to $\mathbf{1}$ and satisfying

$$ \mathcal{R}_{\gamma}(c \in \gamma, u \in [-1, 1]) = \rho_{\alpha(u)}. $$

The homotopy class of $\mathcal{R}_\gamma$ is well-defined.

**Lemma 19.2.** Let $\mathbb{R}P_{\pi}^2$ denote the nonorientable submanifold of $SO(3)$ of the rotations of angle $\pi$. Let $\Sigma$ be a connected surface, orientable or not. Let $f: \partial\Sigma \to SO(3)$ be a map transverse to $\mathbb{R}P_{\pi}^2$. Then $f$ extends to $\Sigma$ if and only if the $\mathbb{Z}/2\mathbb{Z}$-valued algebraic intersection $\langle f(\partial\Sigma), \mathbb{R}P_{\pi}^2\rangle_{SO(3)}$ of $f(\partial\Sigma)$ and $\mathbb{R}P_{\pi}^2$ in $SO(3)$ is zero.

**Remark 19.3.** Recall $\pi_1(SO(3)) = \mathbb{Z}/2\mathbb{Z}$ from Section 4.2. The homotopy class of a loop of $SO(3)$ is determined by its $\mathbb{Z}/2\mathbb{Z}$-valued algebraic intersection with $\mathbb{R}P_{\pi}^2$. This proves Lemma 19.2 when $\Sigma$ is a disk.

**PROOF OF LEMMA 19.2:** When $f$ extends to $\Sigma$, we may choose an extension $f_\Sigma$ transverse to $\mathbb{R}P_{\pi}^2$. Then $f(\partial\Sigma) \cap \mathbb{R}P_{\pi}^2$ bounds the one-manifold $f_\Sigma(\Sigma) \cap \mathbb{R}P_{\pi}^2$. So the $\mathbb{Z}/2\mathbb{Z}$-valued intersection $\langle f(\partial\Sigma), \mathbb{R}P_{\pi}^2\rangle_{SO(3)}$ is zero.

Conversely, assume that $\langle f(\partial\Sigma), \mathbb{R}P_{\pi}^2\rangle_{SO(3)}$ is zero. Then there is a disjoint union $\gamma$ of intervals embedded in $\Sigma$ and transverse to $\partial\Sigma$ with the following properties.

* The boundary $\partial\gamma$ of $\gamma$ is in $\partial\Sigma$.

* The connected components $K$ of $\partial\Sigma$ such that $\langle f(K), \mathbb{R}P_{\pi}^2\rangle = 0$ do not meet $\gamma$

* The connected components $K$ of $\partial\Sigma$ such that $\langle f(K), \mathbb{R}P_{\pi}^2\rangle = 1$ meet $\gamma$ at one point (of $\partial\gamma$).

Then the restriction to $\partial\Sigma$ of the map $\mathcal{R}_\gamma$ associated to $(\Sigma, \gamma)$ as in Notation 19.1 is homotopic to $f$. Therefore, $f$ extends to $\Sigma$. $\square$

Recall that a *framed knot* is a knot equipped with a parallel (up to homotopy). Equivalently, it is a knot equipped with a normal nonzero vector field $\vec{n}_K$. Let $K$ be framed knot in an oriented 3-manifold $M$. Let $\vec{t}_K$ be a tangent vector of $K$ that induces the orientation of $K$. These data induce the direct trivialization $\tau_K$ of $TM|_K$ (up to homotopy) such that $\tau_K(e_1) = \vec{t}_K$ and $\tau_K(e_2) = \vec{n}_K$. The homotopy class of the trivialization $\tau_K$ is well-defined and does not depend on the orientation of $K$.
**Lemma 19.4.** Let $K$ be a framed knot bounding a possibly nonorientable compact surface $\Sigma$ in an oriented 3-manifold $M$. Assume that $\Sigma$ (or, more precisely, the parallel of $K$ on $\Sigma$) induces the framing of $K$. Let $\tau$ be a trivialization of the tangent space of $M$ over $\Sigma$. Then the restriction of $\tau$ to $K$ is not homotopic to $\tau_K$.

**PROOF:** Let us prove that the homotopy class of the restriction of $\tau$ to $K$ is independent of the trivialization $\tau$ of $TM$ over $\Sigma$. Any other trivialization of $TM$ over $\Sigma$ may be written as $\tau \circ \psi_{\mathbb{R}}(f)$ for a map $f$ from $\Sigma$ to $SO(3)$, with the notation of Section 4.2. Lemma 19.2 and Remark 19.3 imply that the restriction $f|_K$ of such a map $f$ is homotopically trivial. Thus, the homotopy class of the restriction of $\tau$ to $K$ is independent of the trivialization $\tau$ of $TM$ over $\Sigma$. It is also independent of the oriented 3-manifold $M$ that contains $\Sigma$. Since the tangent bundle of an oriented 3-manifold over a possibly nonorientable closed surface is trivializable, the homotopy class of the restriction of $\tau$ to $K$ is also independent of $\Sigma$. Therefore, it is enough to prove the lemma when $\Sigma$ is a disk, and it is obvious in this case. $\square$

**Definition 19.5.** If $(K, \tau_K)$ is a framed knot in an oriented 3-manifold $M$ and if $\tau$ is a trivialization of the restriction of $TM$ to $K$, we say that $K$ is $\tau$-bounding if $\tau$ is not homotopic to $\tau_K$. (This notion is independent of the whole manifold $M$, depending only on what happens in a tubular neighborhood of $K$.)

**Definition 19.6.** Let $\Sigma$ be a surface, orientable or not. Let $\gamma$ be a two-sided curve properly embedded in $\Sigma$. Recall the map $\mathcal{R}_\gamma$ from $\Sigma$ to $SO(3)$ from Notation 19.1. Define the *twist map* $\mathcal{T}_\gamma$ *across* $\gamma$ to be the map from $\Sigma \times \mathbb{R}^3$ to itself such that

$$ \mathcal{T}_{\gamma}(y \in \Sigma; X \in \mathbb{R}^3) = (y; \mathcal{R}_{\gamma}(y)(X)). $$

Assume that $\Sigma$ is embedded in an oriented 3-manifold $M$. For a trivialization $\tau$ of $TM|_{\Sigma}$, the *twist of* $\tau|_{\Sigma}$ *across* $\gamma$ is the trivialization $\tau|_{\Sigma} \circ \mathcal{T}_{\gamma}$ (defined up to homotopy).

Let $A$ be a compact oriented connected 3-manifold with boundary $\partial A$. Define the $\mathbb{Z}/2\mathbb{Z}$-Lagrangian $\mathcal{L}_A^{\mathbb{Z}/2\mathbb{Z}}$ of $A$ to be the kernel

$$ \mathcal{L}_A^{\mathbb{Z}/2\mathbb{Z}} = \text{Ker}(H_1(\partial A; \mathbb{Z}/2\mathbb{Z}) \to H_1(A; \mathbb{Z}/2\mathbb{Z})) $$

of the map induced by the inclusion map. This is a Lagrangian subspace of $(H_1(\partial A; \mathbb{Z}/2\mathbb{Z}); \langle \cdot, \cdot \rangle)$.

The curves of an oriented compact surface $F$ embedded in an oriented 3-manifold are naturally framed by the surface $F$: They are framed by a nonzero normal vector field tangent to $F$.
**Proposition 19.7.** Let $\partial A$ be a connected oriented compact surface. Let $\tau$ be a trivialization of $T(\partial A \times [-2, 2])$. Then there exists a unique map

$$ \phi_{\tau}: H_1(\partial A; \mathbb{Z}/2\mathbb{Z}) \to \frac{\mathbb{Z}}{2\mathbb{Z}} $$

such that

1. for a connected curve $x$ of $\partial A = \partial A \times \{0\}$, we have $\phi_{\tau}(x) = 0$ if and only if $x$ (framed by $\partial A$) is $\tau$-bounding and,

2. for any pair $(x, y)$ of elements of $H_1(\partial A; \mathbb{Z}/2\mathbb{Z})$, we have

$$ \phi_{\tau}(x + y) = \phi_{\tau}(x) + \phi_{\tau}(y) + \langle x, y \rangle_{\partial A}. $$

The map $\phi_{\tau}$ satisfies the following properties.

* Let $x$ be a disjoint union of curves in $\partial A$. Assume that $\partial A \times [-2, 2]$ is embedded in an oriented 3-manifold $M$, where $x$ bounds a connected surface $\Sigma$, orientable or not. If $\Sigma$ and $\partial A$ induce the same framing of $x$, then $\tau|_x$ extends to $\Sigma$ as a trivialization of $TM|_{\Sigma}$ if and only if $\phi_{\tau}(x)$ is zero.

* Let $c$ be curve of $\partial A$. Let $\mathcal{T}_c$ denote the twist across $c$ of Definition 19.6. For any $x \in H_1(\partial A; \mathbb{Z}/2\mathbb{Z})$, we have

$$ \phi_{\tau \circ \mathcal{T}_c}(x) = \phi_{\tau}(x) + \langle x, c \rangle_{\partial A}. $$

* When $A$ is a compact oriented connected 3-manifold with boundary $\partial A$, the trivialization $\tau$ extends as a trivialization over $A$ if and only if $\phi_{\tau}(\mathcal{L}_A^{\mathbb{Z}/2\mathbb{Z}}) = \{0\}$.

**PROOF:** For a disjoint union $x = \bigsqcup_{i=1}^n x_i$ of connected curves $x_i$ on $\partial A$, define $\phi_{\tau}(x) = \sum_{i=1}^n \phi_{\tau}(x_i)$ in $\mathbb{Z}/2\mathbb{Z}$, with $\phi_{\tau}(x_i) = 0$ if $x_i$ (framed by $\partial A$) is $\tau$-bounding, and $\phi_{\tau}(x_i) = 1$ otherwise.

Assume that such a framed disjoint union $x$ bounds a connected surface $\Sigma$ in an oriented 3-manifold $M$. Also assume that $\Sigma$ and $\partial A$ induce the same framing of $x$. Let us prove that $\tau|_x$ extends to $\Sigma$ as a trivialization of $TM|_{\Sigma}$ if and only if $\phi_{\tau}(x)$ is zero.

When $\phi_{\tau}(x) = 0$, group all the curves $x_i$ such that $\phi_{\tau}(x_i) = 1$ by pairs. Make each such pair bound an annulus that induces the framing. Make each curve $x_i$ such that $\phi_{\tau}(x_i) = 0$ bound a disk which induces the framing. Let $\hat{\Sigma}$ denote the union of $\Sigma$ with the above disks and annuli. The restriction to $\hat{\Sigma}$ of the tangent bundle of an oriented 3-manifold $M$ in which $\hat{\Sigma}$ embeds is
independent of $M$. It admits a trivialization $\hat{\tau}$. Extend $\tau$ to $\hat{\Sigma} \setminus \hat{\Sigma}$. Write the restriction of $\hat{\tau}$ to $(\hat{\Sigma} \setminus \hat{\Sigma}) \times \mathbb{R}^3$ as $\hat{\tau} = \tau \circ \psi_{\mathbb{R}}(f)$ for a map $f$ from $\hat{\Sigma} \setminus \hat{\Sigma}$ to $\text{SO}(3)$. According to Lemma 19.2, the intersection of $f(x)$ and $\mathbb{R}P_{\pi}^2$ is zero, and the map $f$ extends to $\hat{\Sigma}$. Thus $\tau = \hat{\tau} \circ \psi_{\mathbb{R}}(f)^{-1}$ also extends to $\Sigma$.

When $\phi_{\tau}(x) = 1$, assume that $\phi_{\tau}(x_1) = 1$ without loss of generality. Group the other curves $x_i$ such that $\phi_{\tau}(x_i) = 1$ pairwise, make them cobound a disjoint union of annuli, and make the curves $x_i$ such that $\phi_{\tau}(x_i) = 0$ bound disks, in a framed way as above. Let $\hat{\Sigma}$ denote the union of $\Sigma$ with these disks and these annuli. The boundary of $\hat{\Sigma}$ is $x_1$. The trivialization $\tau$ still extends to $\hat{\Sigma} \setminus \hat{\Sigma}$. If $\tau$ extends to $\Sigma$, then it extends to $\hat{\Sigma}$, and $x_1$ is $\tau$-bounding. So $\phi_{\tau}(x_1) = 0$, which is absurd. Therefore, $\tau$ does not extend to $\Sigma$.

Let us prove that our above definition of $\phi_{\tau}(x)$ depends only on the class of $x$ in $H_1(\partial A; \mathbb{Z}/2\mathbb{Z})$. Let $x$ be an embedded (possibly nonconnected) curve in $\partial A$. Let $y$ be another such curve in $\partial A \times \{-1\}$ homologous to $x$ modulo $2H_1(\partial A; \mathbb{Z})$. Then there exists a framed (possibly nonorientable) cobordism between $x$ and $y$ in $\partial A \times [-1, 1]$, and it is easy to see that $\phi_{\tau}(x) = 0$ if and only if $\phi_{\tau}(y) = 0$.

Let us check that $\phi_{\tau}$ behaves as predicted under addition. Because we are dealing with elements of $H_1(\partial A; \mathbb{Z}/2\mathbb{Z})$, we can consider representatives $x$ and $y$ of $x$ and $y$ such that $x$ is connected and intersects $y$ at most once. Next, the known additivity under disjoint union reduces the proof to the case in which $x$ and $y$ are connected and $x$ and $y$ intersect once. Note that both sides of the equality to be proved vary in the same way under trivialization changes. Consider the punctured torus neighborhood of $x \cup y$ and a trivialization $\tau$ that restricts to the punctured torus as the direct sum of a parallelization of the torus and the normal vector to $\partial A$. Then we have $\phi_{\tau}(x+y) = \phi_{\tau}(x) = \phi_{\tau}(y) = 1$. We leave the last two assertions to the reader. $\square$

**Example 19.8.** For any $\mathbb{Q}$-handlebody $A$, there exists a Lagrangian subspace $\mathcal{L}^{\mathbb{Z}}$ of $(H_1(\partial A; \mathbb{Z}); \langle . , . \rangle)$, such that $\mathcal{L}_A = \mathcal{L}^{\mathbb{Z}} \otimes \mathbb{Q}$. The following example shows that $\mathcal{L}_A^{\mathbb{Z}/2\mathbb{Z}}$ may differ from $\mathcal{L}^{\mathbb{Z}} \otimes \mathbb{Z}/2\mathbb{Z}$.

Let $\mathcal{M}$ be a Möbius band embedded in the interior of a solid torus $D^2 \times S^1$ so that the core of the solid torus is the core of $\mathcal{M}$. Embed $D^2 \times S^1$ into $S^2 \times S^1 = D^2 \times S^1 \cup_{\partial D^2 \times S^1} (-D^2 \times S^1)$ as the first copy. Orient the knot $\partial \mathcal{M}$ so that $\partial \mathcal{M}$ pierces twice $S^2 \times 1$ positively. Let $m$ be the meridian of $\partial \mathcal{M}$. Let $\ell$ be the parallel of $\partial \mathcal{M}$ induced by $\mathcal{M}$. Let $A$ be the exterior of the knot $\partial \mathcal{M}$ in $S^2 \times S^1$. The reader can check that $A$ is a $\mathbb{Q}$-handlebody, as an exercise. Observe $\mathcal{L}_A^{\mathbb{Z}} = \mathbb{Z}[2m]$, $\mathcal{L}_A = \mathbb{Q}[m]$, and $\mathcal{L}_A^{\mathbb{Z}/2\mathbb{Z}} = \mathbb{Z}/2\mathbb{Z}[\ell]$.

Let $A_0$ be the solid torus with boundary $\partial A = \partial A_0$, where $m$ bounds a disk. We have $\mathcal{L}_{A_0} = \mathbb{Q}[m]$. Equip $A_0$ with a parallelization $\tau$ such that
$\phi_{\tau}(\ell) = 1$. According to Proposition 19.7, the restriction to $\partial A$ of $\tau$ does not extend to $A$.

## 19.2 Definition of pseudo-parallelizations

**Definition 19.9.** Recall $\varepsilon$ and the maps $\alpha$ and $\rho_{\alpha(u)}$ from Notation 19.1. Set $N(\partial[-1, 1]) = [-1, 1] \setminus \dots -1 + \varepsilon, 1 - \varepsilon$. A pseudo-parallelization $\tilde{\tau} =$ ($N(\gamma)$; $\tau_e, \tau_b$) of an oriented 3-manifold $A$ with possible boundary consists of

* a framed link $\gamma$ of the interior of $A$, which will be called the link of the pseudo-parallelization $\tilde{\tau}$, equipped with a neighborhood $N(\gamma) = [a, b] \times \gamma \times [-1, 1]$, for real numbers $a$ and $b$ such that $a < b$ and $\varepsilon < \frac{b-a}{4}$,

* a parallelization $\tau_e$ of $A$ outside $[a + \varepsilon, b - \varepsilon] \times [\gamma \times [-1 + \varepsilon, 1 - \varepsilon[$,

* a parallelization $\tau_b: N(\gamma) \times \mathbb{R}^3 \to TN(\gamma)$ of $N(\gamma)$ such that

$$ \tau_b = \begin{cases} \tau_e & \text{over } [b - \varepsilon, b] \times \gamma \times [-1, 1] \text{ and } [a, b] \times \gamma \times N(\partial[-1, 1]) \\ \tau_e \circ \mathcal{T}_\gamma & \text{over } [a, a + \varepsilon] \times \gamma \times [-1, 1], \end{cases} $$

where

$$ \mathcal{T}_{\gamma}(t, c \in \gamma, u \in [-1, 1]; X \in \mathbb{R}^3) = (t, c, u; \rho_{\alpha(u)}(X)). $$

**Lemma 19.10.** Let $A$ be a compact oriented 3-manifold and let $\tau$ be a trivialization of $TA$ defined on a collar $[-4, 0] \times \partial A$ of $\partial A(= \{0\} \times \partial A)$. Then there is a pseudo-parallelization of $A$ that extends the restriction of $\tau$ to $[-1, 0] \times \partial A$.

**PROOF:** There exists a trivialization $\tau'$ of $TA$ on $A$. After a homotopy of $\tau$ around $\{-2\} \times \partial A$, there exists a union $\gamma \times [-1, 1]$ of annuli of $\{-2\} \times \partial A$ such that $\tau = \tau' \circ \mathcal{T}_{\gamma}$ on $\{-2\} \times \partial A$. (When $\partial A$ is connected, $\gamma$ can be assumed to be connected, too.) Consider the neighborhood $N(\gamma) = [-2, -1] \times \gamma \times [-1, 1]$ of $\gamma$. Define $\tau_e$ to coincide with $\tau$ on $([-2, 0] \times \partial A) \setminus \text{Int}(N(\gamma))$, and with $\tau'$ on $A \setminus ([]-2, 0] \times \partial A)$. Define $\tau_b$ to coincide with $\tau$ on $N(\gamma)$. $\square$

**Definition 19.11.** [Trivialization $\tilde{\tau}_C$ of $TA \otimes_R \mathbb{C}$] Define a smooth map

$$ F_U: [a, b] \times [-1, 1] \to SU(3) $$

such that

$$ F_U(t, u) = \begin{cases} \mathbf{1} & \text{if } |u| > 1 - \varepsilon \\ \rho_{\alpha(u)} & \text{if } t < a + \varepsilon \\ \mathbf{1} & \text{if } t > b - \varepsilon. \end{cases} $$
Since $\pi_1(SU(3))$ is trivial, it is possible to define such a smooth map. Define a trivialization $\tilde{\tau}_C$ of $TA \otimes_R C$, associated with the pseudo-parallelization $\tilde{\tau}$ of Definition 19.9, as follows.

* On $(A \setminus N(\gamma)) \times \mathbb{C}^3$, we have $\tilde{\tau}_C = \tau_e \otimes 1_C$,

* Over $[a, b] \times \gamma \times [-1, 1]$, we have $\tilde{\tau}_C(t, c, u; X) = \tau_b(t, c, u; F_U(t, u)^{-1}(X))$.

Since $\pi_2(SU(3))$ is trivial, the homotopy class of $\tilde{\tau}_C$ is well-defined.

**Definition 19.12.** Let $M_0$ and $M_1$ be two compact connected oriented 3-manifolds whose boundaries $\partial M_0$ and $\partial M_1$ have collars identified by a diffeomorphism. Let $\tau_0$ be a pseudo-parallelization of $M_0$, which restricts to a collar neighborhood of $\partial M_0$ as a genuine trivialization. Let $\tau_1$ be a pseudo-parallelization of $M_1$ that coincides with $\tau_0$ on this collar neighborhood. We use the definition of Proposition 5.10 of relative Pontrjagin numbers and define $p_1(\tau_0, \tau_1)$ to be $p_1(\tau_{0,C}, \tau_{1,C})$. Let $\tilde{R}$ be a rational homology $\mathbb{R}^3$. A pseudo-parallelization $\tilde{\tau}$ of $\tilde{R}$ is asymptotically standard if it coincides with the standard parallelization $\tau_s$ of $\mathbb{R}^3$ outside $B_R$. (Recall Definition 3.6.) For such an asymptotically standard pseudo-parallelization, set $p_1(\tilde{\tau}) = p_1((\tau_s)|_{B^3}, \tilde{\tau}|_{B_R})$.

**Definition 19.13.** [Homogeneous boundary form associated with $\tilde{\tau}$] Let $\tilde{\tau} = (N(\gamma); \tau_e, \tau_b)$ be a pseudo-parallelization of a 3-manifold $A$. Recall $\varepsilon$ and the map $\alpha$ from Notation 19.1 and Definition 19.9. Define a smooth map

$$F: [a, b] \times [-1, 1] \to SO(3)$$

such that

$$F(t, u) = \begin{cases} 1 & \text{if } |u| > 1 - \varepsilon \\ \rho_{\alpha(u)} & \text{if } t < a + \varepsilon \\ \rho_{-\alpha(u)} & \text{if } t > b - \varepsilon. \end{cases}$$

Since the restriction of $F$ to the boundary of $[a, b] \times [-1, 1]$ is trivial in $\pi_1(SO(3))$, it is possible to define such a smooth map $F$.

Let $p_{\tau_b} = p(\tau_b)$ denote the projection from $UN(\gamma)$ to $S^2$ induced by $\tau_b$. We have $p_{\tau_b}(\tau_b(t, c, u; X \in S^2)) = X$. Define

$$F(\gamma, \tau_b): \begin{cases} [a, b] \times \gamma \times [-1, 1] \times S^2 \\ (t, c, u; Y) \end{cases} \to \begin{cases} [a, b] \times \gamma \times [-1, 1] \times S^2 \\ (t, c, u; F(t, u)(Y)) \end{cases}.$$

Define the closed two-form $\omega(\gamma, \tau_b)$ on $U([a, b] \times \gamma \times [-1, 1])$ to be

$$\omega(\gamma, \tau_b) = \frac{p(\tau_b \circ \mathcal{T}_{\gamma}^{-1})^*(\omega_{S^2}) + p(\tau_b \circ F(\gamma, \tau_b)^{-1})^*(\omega_{S^2})}{2}.$$
The homogeneous boundary form associated to $(\tilde{\tau}, F)$ is the following closed 2-form $\omega(\tilde{\tau}, F)$ on $UA$.

$$ \omega(\tilde{\tau}, F) = \begin{cases} p_{\tau_e}^* (\omega_{S^2}) & \text{on } U(A \setminus N(\gamma)) \\ \omega(\gamma, \tau_b) & \text{on } U(N(\gamma)). \end{cases} $$

A homogeneous boundary form of $(UA, \tilde{\tau})$ is a homogeneous boundary form associated to $(\tilde{\tau}, F)$ for some $F$ as above.

We will justify the consistency of Definition 19.13 by applying the following lemma with the constant map $\kappa$ with value one. We will use the general lemma in Lemma 21.3.

**Lemma 19.14.** Let $(e_1 = (1, 0, 0), e_2 = (0, 1, 0), e_3 = (0, 0, 1))$ denote the standard basis of $\mathbb{R}^3$. Let $v_i : \mathbb{R}^3 \to \mathbb{R}$ denote the ith coordinate with respect to this basis. Let $\rho_\theta = \rho_{\theta,e_3}$ denote the rotation of $\mathbb{R}^3$ of angle $\theta$ whose axis is directed by $e_3$. For $k \in \mathbb{Z}$, define

$$ \mathcal{T}_k : \begin{array}{ccc} \mathbb{R} \times S^2 & \to & S^2 \\ (\theta, X) & \mapsto & \rho_{k\theta}(X). \end{array} $$

Let $\kappa: [-1, 1] \to \mathbb{R}$ be a smooth map. Consider the associated volume form $(\kappa \circ v_3)\omega_{S^2}$ on $S^2$. It is invariant under the rotations $\rho_\theta$. Then we have

$$ \mathcal{T}_k^* ((\kappa \circ v_3)\omega_{S^2}) = \mathcal{T}_0^* ((\kappa \circ v_3)\omega_{S^2}) + \frac{k(\kappa \circ v_3)}{4\pi} d\theta \wedge dv_3 $$

**PROOF:** Recall the homogeneous two-form $\omega_{S^2}$ on $S^2$ with total area 1. When $X \in S^2$, and when $v$ and $w$ are two tangent vectors of $S^2$ at $X$, we have

$$ \omega_{S^2}(v \wedge w) = \frac{1}{4\pi} \det(X, v, w), $$

where $X \wedge v \wedge w = \det(X, v, w)e_1 \wedge e_2 \wedge e_3$ in $\Lambda^3 \mathbb{R}^3$.

Since $\rho_\theta$ preserves the area in $S^2$ and leaves $v_3$ invariant, the restrictions of $\mathcal{T}_k^* ((\kappa \circ v_3)\omega_{S^2})$ and $\mathcal{T}_0^* ((\kappa \circ v_3)\omega_{S^2})$ coincide on $\Lambda^2 T_{(\theta,X)}(\{\theta\} \times S^2)$. Therefore, we are left with the computation of

$$ (\mathcal{T}_k^* ((\kappa \circ v_3)\omega_{S^2}) - \mathcal{T}_0^* ((\kappa \circ v_3)\omega_{S^2})) (u \wedge v) $$

when $u \in T_{(\theta,X)}(\mathbb{R} \times \{X\})$ and $v \in T_{(\theta,X)}(\{\theta\} \times S^2)$. We have

$$ \mathcal{T}_0^* ((\kappa \circ v_3)\omega_{S^2}) (u \wedge v) = 0, $$

$$ \mathcal{T}_k^* ((\kappa \circ v_3)\omega_{S^2})_{(\theta,X)} (u \wedge v) = \frac{\kappa \circ v_3(X)}{4\pi} \det(\rho_{k\theta}(X), T_{(\theta,X)}\mathcal{T}_k(u), T_{(\theta,X)}\mathcal{T}_k(v)), $$
and $T_{(\theta,X)}\mathcal{T}_k(v) = \rho_{k\theta}(v)$. Since $\rho_{k\theta}$ preserves the volume in $\mathbb{R}^3$, we get

$$ \mathcal{T}_k^*((\kappa \circ v_3)\omega_{S^2})_{(\theta,X)}(u \wedge v) = \frac{\kappa \circ v_3(X)}{4\pi} \det(X, \rho_{-k\theta}(T_{(\theta,X)}\mathcal{T}_k(u)), v). $$

Let $X_i$ stand for $v_i(X)$. We have

$$ T_{(\theta,X)}\mathcal{T}_k(u) = kd\theta(u)\rho_{k\theta+\pi/2}(X_1e_1 + X_2e_2). $$

We obtain

$$ \mathcal{T}_k^*(\omega_{S^2})_{(\theta,X)}(u \wedge v) = \frac{kd\theta(u)}{4\pi} \det(X, -X_2e_1 + X_1e_2, v), $$

and therefore

$$ \begin{aligned} \mathcal{T}_k^*(\omega_{S^2})(u \wedge .) &= \frac{kd\theta(u)}{4\pi} \det \begin{pmatrix} X_1 & -X_2 & dv_1 \\ X_2 & X_1 & dv_2 \\ X_3 & 0 & dv_3 \end{pmatrix} \\ &= \frac{kd\theta(u)}{4\pi} (-X_3X_1dv_1 - X_3X_2dv_2 + (1-X_3^2)dv_3) \\ &= \frac{kd\theta(u)}{4\pi} dv_3. \end{aligned} $$

□

**PROOF OF THE CONSISTENCY OF DEFINITION 19.13:** It suffices to prove

$$ p(\tau_b \circ \mathcal{T}_{\gamma}^{-1})^* (\omega_{S^2}) + p(\tau_b \circ \mathcal{T}_{\gamma})^* (\omega_{S^2}) = 2p(\tau_b)^* (\omega_{S^2}) $$

on $U([b - \varepsilon, b] \times \gamma \times [-1, 1])$, where we have

$$ p_{\tau_b \circ \mathcal{T}_{\gamma}^{\pm 1}}(\tau_b(t, c, u; X)) = p_{\tau_b \circ \mathcal{T}_{\gamma}^{\pm 1}}(\tau_b \circ \mathcal{T}_{\gamma}^{\pm 1}(t, c, u; \rho_{\mp\alpha}(u)(X))) = \rho_{\mp\alpha(u)}(X). $$

Set $\tilde{p}_{\tau_b} = p_{[-1,1]} \times p_{\tau_b}: U([b - \varepsilon, b] \times \gamma \times [-1, 1]) \to [-1, 1] \times S^2$. We have

$$ p(\tau_b) = \mathcal{T}_0 \circ (\alpha \times \mathbf{1}_{S^2}) \circ \tilde{p}_{\tau_b} \text{ and } p(\tau_b \circ \mathcal{T}_{\gamma}^{\pm 1}) = \mathcal{T}_{\mp 1} \circ (\alpha \times \mathbf{1}_{S^2}) \circ \tilde{p}_{\tau_b}. $$

We get $p(\tau_b \circ \mathcal{T}_{\gamma}^{\pm 1})^* (\omega_{S^2}) = ((\alpha \times \mathbf{1}_{S^2}) \circ \tilde{p}_{\tau_b})^* (\mathcal{T}_{\mp 1}^*(\omega_{S^2}))$.

Thus, Lemma 19.14 implies that Definition 19.13 is consistent. □

**Definition 19.15.** Let $\tilde{R}$ be a rational homology $\mathbb{R}^3$, equipped with an asymptotically standard pseudo-parallelization $\tilde{\tau}$. A homogeneous propagating form of $(C_2(R), \tilde{\tau})$ is a propagating form of $C_2(R)$ (as in Definition 3.11) that coincides with a homogeneous boundary form of $(U\tilde{R}, \tilde{\tau})$ as in Definition 19.13 on $U\tilde{R}$.
**Lemma 19.16.** *Such homogeneous propagating forms exist for any $(R, \tilde{\tau})$.*

**PROOF:** See Section 3.3. □

<figure><img src="image_154.png" /></figure>

As in Definition 12.4, a *pseudo-parallelization* of a rational homology
cylinder $\mathcal{C}$ is a pseudo-parallelization of $\tilde{R}(\mathcal{C})$ that agrees with the standard
parallelization of $\mathbb{R}^3$ outside $\mathcal{C}$.

The main result of this chapter is the following theorem.

**Theorem 19.17.** Let $\mathcal{C}$ be a rational homology cylinder. Let $\tau = (N(\gamma); \tau_e, \tau_b)$ be a pseudo-parallelization of $\mathcal{C}$. Let $L: \mathcal{L} \hookrightarrow \tilde{R}(\mathcal{C}) \setminus N(\gamma)$ be a long tangle representative in $\tilde{R}(\mathcal{C}) \setminus N(\gamma)$.

*Definition 12.6 of $I_\theta(K, \tau)$ naturally extends for such a pseudo-parallelization* $\tau$ when $K$ is a component of $L$.

With this extended definition of $I_\theta$ and with Definition 19.12 of $p_1(\tau)$,
Theorem 7.20 and Theorem 12.7 also hold when $\tau$ is a pseudo-parallelization
$\tau = (N(\gamma); \tau_e, \tau_b)$ of $\mathcal{C}$ such that $N(\gamma)$ does not meet the image of the long
tangle representative (or the link) $L: \mathcal{L} \hookrightarrow \tilde{R}(\mathcal{C})$.

In order to prove Theorem 19.17 in Section 19.5, we prove some prelimi-
nary lemmas in the next sections.

**19.3 Integration of homogeneous propagating forms along surfaces**

**Definition 19.18.** Let $\Sigma$ be a compact oriented surface with boundary. Let $\Sigma_0$ denote the image of the zero section in the tangent bundle $T\Sigma$ of $\Sigma$. Let $X$ be a nowhere vanishing section of $T\Sigma$ along the boundary of $\Sigma$. Let $\tilde{X}$ be an extension of $X$ over $\Sigma$ whose image $\tilde{X}(\Sigma)$ in $T\Sigma$ is transverse to $\Sigma_0$. The sections $\Sigma_0$ and $\tilde{X}(\Sigma)$ are naturally oriented by $\Sigma$. The relative Euler number $\chi(X; \Sigma)$ is their algebraic intersection $\langle \tilde{X}(\Sigma), \Sigma_0 \rangle_{T\Sigma}$ in $T\Sigma$.

Note that this definition makes sense since all the extensions of $X$ are
homotopic relatively to $\partial\Sigma$. This Euler number is an obstruction to extending
$X$ over $\Sigma$ as a nowhere vanishing section of $T\Sigma$. Here are some other well-
known properties of this number.

**Lemma 19.19.** Let $\Sigma$ be a compact oriented surface with boundary, and let $X$ be a section of $U\Sigma$ along the boundary of $\Sigma$.

* If $\Sigma$ is connected and if $\chi(X; \Sigma) = 0$, then $X$ extends as a nowhere vanishing section of $T\Sigma$.
* If $X$ is tangent to the boundary of $\Sigma$, then $\chi(X; \Sigma)$ is the Euler characteristic $\chi(\Sigma)$ of $\Sigma$.

* More generally, let $a^{(1)}, \dots, a^{(k)}$ denote the $k$ connected components of the boundary $\partial\Sigma$ of $\Sigma$. For $i = 1, \dots, k$, the unit bundle $U\Sigma|_{a^{(i)}}$ of $T\Sigma|_{a^{(i)}}$ is an $S^1$-bundle over $a^{(i)}$ with a canonical trivialization induced by $Ta^{(i)}$. Let $d(X, a^{(i)})$ be the degree of the projection on the fiber $S^1$ of this bundle of the section $X$, with respect to this canonical trivialization. Then we have

$$ \chi(X; \Sigma) = \sum_{i=1}^{k} d(X, a^{(i)}) + \chi(\Sigma). $$

**PROOF:** First observe all these properties when $\Sigma$ is a disk. When $\Sigma$ is connected, there is a disk $D$ that contains all the zeros of an arbitrary generic extension of $\tilde{X}$ of $X$. If $\chi(X; \Sigma) = 0$, then $\chi(\tilde{X}|_{\partial D}; D) = 0$, and $\tilde{X}|_{\partial D}$ extends to $D$ as a nowhere vanishing section. So $X$ extends to $\Sigma$ as a nowhere vanishing section.

Let $U^+\partial\Sigma$ denote the unit vector field of $\partial\Sigma$ that is tangent to $\partial\Sigma$ and induces its orientation. Let us prove that the following equality $(\ast(\Sigma))$

$$ \chi(U^+\partial\Sigma; \Sigma) = \chi(\Sigma) $$

holds for a general $\Sigma$.

For $i = 1, 2$, let $\Sigma_i$ be a compact oriented surface, and let $c_i$ be a connected component of $\partial\Sigma_i$. Set $\Sigma = \Sigma_1 \cup_{c_1\sim-c_2} \Sigma_2$. Since the section $U^+c_1$ is homotopic to $(-U^+c_1)$ as a section of $U\Sigma$, we have

$$ \chi(U^+\partial\Sigma; \Sigma) = \chi(U^+\partial\Sigma_1; \Sigma_1) + \chi(U^+\partial\Sigma_2; \Sigma_2). $$

Assume that $(\ast(\Sigma_1))$ holds. Since $\chi(\Sigma) = \chi(\Sigma_1) + \chi(\Sigma_2)$, the equalities $(\ast(\Sigma_2))$ and $(\ast(\Sigma))$ are equivalent. Since $S^1 \times S^1$ is parallelizable, the equality $(\ast(S^1 \times S^1))$ holds. So $(\ast(S^1 \times S^1 \setminus \tilde{D}^2))$ holds. The general case follows easily. The third property of $\chi(X; \Sigma)$ is an easy consequence of the previous one. $\square$

**Lemma 19.20.** Recall the vectors $e_2 = (0, 1, 0)$ and $e_3 = (0, 0, 1)$ of $\mathbb{R}^3$. Let $\Sigma$ be a compact oriented surface immersed in a 3-manifold $M$ equipped with a parallelization $\tau$. Assume that $\tau(\cdot \times e_3)$ is a positive normal to $\Sigma$ along $\partial\Sigma$. Let $s_+(\Sigma) \subset UM$ (resp. $s_-(\Sigma) \subset UM$) be the graph of the section of $UM|_{\Sigma}$ in $UM$ associated to the positive (resp. negative) normal to $\Sigma$. Let $s_\tau(\Sigma; e_3)$ be the graph of the section $\tau(\Sigma \times \{e_3\})$. Then the cycles

$$ 2(s_+(\Sigma) - s_\tau(\Sigma; e_3)) - \chi(\tau(\cdot \times e_2)|_{\partial\Sigma}; \Sigma)UM|_* $$
and

$$2(s_{-}(\Sigma) - s_{\tau}(\Sigma; -e_{3})) + \chi(\tau(\cdot \times e_{2})|_{\partial\Sigma}; \Sigma)UM|_{*}$$

of $UM|_{\Sigma}$ are null-homologous in $UM|_{\Sigma}$.

**PROOF:** Consider the involution $\iota_{\Sigma}$ of $UM|_{\Sigma}$ that sends a vector to its opposite. This involution reverses the orientation of a fiber $UM|_{*}$. Therefore, it sends the second cycle (with $s_{-}(\Sigma)$) to the first one (with $s_{+}(\Sigma)$). Thus, it suffices to prove that

$$2(s_{+}(\Sigma) - s_{\tau}(\Sigma; e_{3})) - \chi(\tau(\cdot \times e_{2})|_{\partial\Sigma}; \Sigma)UM|_{*}$$

is a null-homologous cycle in $UM|_{\Sigma}$. The trivialization $\tau$ can be homotoped so that $\tau(\cdot \times e_{3})$ is a positive normal to $\Sigma$ over a one-skeleton of $\Sigma$. Therefore, we can assume that $\tau(\cdot \times e_{3})$ is a positive normal to $\Sigma$ over the complement of a disjoint union of disks embedded in the interior of $\Sigma$. So it suffices to prove that $2(s_{+}(D) - s_{\tau}(D; e_{3})) - \chi(\tau(\cdot \times e_{2})|_{\partial D}; D)UM|_{*}$ is null-homologous, for such a disk $D$. Let $\tau_{0}$ be a trivialization of $UM|_{D}$ such that $\tau_{0}(\cdot, e_{3})$ is the positive normal to $D$ and $\tau_{0}(\cdot, e_{2})$ is tangent to $\Sigma$. The trivialization $\tau_{0}$ identifies $UM|_{D}$ with $D \times S^{2}$, and we have

$$[s_{+}(D)] - [s_{\tau}(D; e_{3})] = -\langle s_{\tau}(D; e_{3}), s_{\tau_{0}}(D; -e_{3}) \rangle_{UM|_{D}} [UM|_{*}].$$

Let $[e_{2}, e_{3}]$ denote the shortest arc of great circle from $e_{2}$ to $e_{3}$ on $S^{2}$. We have $\langle s_{\tau}(D; e_{3}), s_{\tau_{0}}(D; -e_{3}) \rangle_{UM|_{D}} = \langle s_{\tau}(D; e_{2}), s_{\tau_{0}}(D; -e_{3}) \rangle_{UM|_{D}}$ because $s_{\tau}(D; e_{3}) \cap s_{\tau_{0}}(D; -e_{3})$ and $s_{\tau}(D; e_{2}) \cap s_{\tau_{0}}(D; -e_{3})$ cobound $\tau(D \times [e_{2}, e_{3}]) \cap s_{\tau_{0}}(D; -e_{3})$ in the interior of $UM|_{D}$. We similarly get

$$\langle s_{\tau}(D; e_{3}), s_{\tau_{0}}(D; -e_{3}) \rangle_{UM|_{D}} = \langle s_{\tau}(D; -e_{2}), s_{\tau_{0}}(D; -e_{3}) \rangle_{UM|_{D}}.$$

Applying the involution $\iota_{\Sigma}$, we obtain

$$2([s_{+}(D)] - [s_{\tau}(D; e_{3})]) = \langle s_{\tau}(D; e_{2}), s_{\tau_{0}}(D; e_{3}) - s_{\tau_{0}}(D; -e_{3}) \rangle_{UM|_{D}} [UM|_{*}].$$

The projection to $\mathbb{R}^{2} \times \{0\}$ of $\tau(\cdot \times e_{2})$ is an extension of the section $\tau(\cdot \times e_{2})$ of $TD$ to $D$. Its intersection with the zero section of $TD$ is the above intersection number. $\square$

**Definition 19.21.** A homotopy from a pseudo-parallelization $(N(\gamma); \tau_{e}, \tau_{b})$ to another such $(N(\gamma); \tau'_{e}, \tau'_{b})$ is a homotopy from the pair $(\tau_{e}, \tau_{b})$ to the pair $(\tau'_{e}, \tau'_{b})$ such that

$$\tau_{b} = \begin{cases} \tau_{e} & \text{over } \partial([a, b] \times \gamma \times [-1, 1]) \setminus (\{a\} \times \gamma \times [-1, 1]) \\ \tau_{e} \circ \mathcal{T}_{\gamma} & \text{over } \{a\} \times \gamma \times [-1, 1] \end{cases}$$

at any time.
**Proposition 19.22.** Let $\Sigma$, $e_3$, $s_+(\Sigma)$, and $s_-(\Sigma)$ be as in Lemma 19.20. Let $\tilde{\tau}$ be a pseudo-parallelization restricting to a neighborhood of $\partial\Sigma$ as a genuine parallelization of $M$ such that $\tilde{\tau}(\cdot \times e_3)$ is the positive normal to $\Sigma$ along $\partial\Sigma$. Let $\omega(\tilde{\tau})$ be a homogeneous boundary form associated with $\tilde{\tau}$ (as in Definition 19.13). Then we have

$$ \int_{s_+(\Sigma)} \omega(\tilde{\tau}) = \frac{1}{2} \chi(\tilde{\tau}(\cdot \times e_2)|_{\partial\Sigma}; \Sigma) $$

and

$$ \int_{s_-(\Sigma)} \omega(\tilde{\tau}) = -\frac{1}{2} \chi(\tilde{\tau}(\cdot \times e_2)|_{\partial\Sigma}; \Sigma). $$

**PROOF:** Observe that $\omega(\tilde{\tau})$ is a closed form, which satisfies $\int_{UM|_*} \omega(\tilde{\tau}) = 1$. So Lemma 19.20 implies

$$ \int_{s_{\pm}(\Sigma)} \omega(\tilde{\tau}) = \mp \frac{1}{2} \chi(\tilde{\tau}(\cdot \times e_2)|_{\partial\Sigma}; \Sigma) + \int_{s_{\tilde{\tau}}(\Sigma; \pm e_3)} \omega(\tilde{\tau}). $$

When $\tilde{\tau}$ is a genuine parallelization, the term $\int_{s_{\tilde{\tau}}(\Sigma; \pm e_3)} \omega(\tilde{\tau})$ is zero, and the proposition follows.

In general, for $\tilde{\tau} = (N(\gamma) = [a, b] \times \gamma \times [-1, 1]; \tau_e, \tau_b)$, the integral $\int_{s_+(\Sigma)} \omega(\tilde{\tau})$ is invariant under an isotopy of $\Sigma$ that fixes $\partial\Sigma$ since $\omega(\tilde{\tau})$ is closed. It is also invariant under a homotopy of $(\tau_e, \tau_b)$ as in Definition 19.21 that is fixed in a neighborhood of $\partial\Sigma$. (See Lemma B.2.)

In particular, there is no loss of generality in assuming that $\Sigma$ meets $N(\gamma)$ along disks $D_c = [a, b] \times \{c\} \times [-1, 1]$, and that $\tau_b(\cdot \times e_3)$ is the positive normal to $D_c$ along $\partial D_c$ for these disks. Thus, thanks to the good behavior of the two sides of the equality to be proved under gluings along circles that satisfy the boundary conditions, it suffices to prove the proposition when $\Sigma$ is a meridian disk $D_c$ of $\gamma$ (with its corners smoothed) such that $\tau_b(\cdot \times e_3)$ is the positive normal to $\Sigma$ along $\partial\Sigma$. On $UM|_{D_c}$, the form $\omega(\tilde{\tau})$ is then equal to

$$ \omega(\gamma, \tau_b) = \frac{p(\tau_b \circ \mathcal{T}_{\gamma}^{-1})^* (\omega_{S^2}) + p(\tau_b \circ F(\gamma, \tau_b)^{-1})^* (\omega_{S^2})}{2}, $$

where $p(\tau_b \circ \mathcal{T}_{\gamma}^{-1})^* (\omega_{S^2})$ and $p(\tau_b \circ F(\gamma, \tau_b)^{-1})^* (\omega_{S^2})$ are propagating forms respectively associated with the parallelizations $\tau_b \circ \mathcal{T}_{\gamma}^{-1}$ and $(\tau_b \circ F(\gamma, \tau_b)^{-1})$, and $\chi(\tau_b \circ \mathcal{T}_{\gamma}^{-1}(\cdot \times e_2)|_{\partial\Sigma}; \Sigma) = \chi(\tau_b(\cdot \times e_2)|_{\partial\Sigma}; \Sigma)$.

Therefore, we have

$$ \int_{s_+(\Sigma)} \omega(\tilde{\tau}) = \frac{1}{4} \left( \chi(\tau_b(\cdot \times e_2)|_{\partial\Sigma}; \Sigma) + \chi((\tau_b \circ F(\gamma, \tau_b)^{-1})(\cdot \times e_2)|_{\partial\Sigma}; \Sigma) \right). $$
Thanks to Lemma 19.19, this average is $\frac{1}{2}(\chi(\tau_e(\cdot \times e_2)|_{\partial\Sigma}; \Sigma))$. This con- cludes the computation of $\int_{s_+(\Sigma)} \omega(\tilde{\tau})$. The computation of $\int_{s_-(\Sigma)} \omega(\tilde{\tau})$ is similar. $\square$

## 19.4 Anomalous terms for pseudo-parallelizations

**Proposition 19.23.** Let A be a compact 3-manifold equipped with two pseudo-parallelizations $\tau_0$ and $\tau_1$ that coincide with a common genuine parallelization along a regular neighborhood of $\partial A$. There exists a closed 2-form $\omega$ on $[0, 1] \times UA$ that restricts

* to $\{0\} \times UA$ as a homogeneous boundary form $\omega(\tau_0)$ of $(UA, \tau_0)$,

* to $\{1\} \times UA$ as a homogeneous boundary form $\omega(\tau_1)$ of $(UA, \tau_1)$,

* to $[0, 1] \times UA|_{\partial A}$ as $p_{UA}^*(\omega(\tau_0))$ with respect to the natural projection $p_{UA}: [0, 1] \times UA \to UA$.

**PROOF:** Without loss of generality, assume that A is connected. Set $X = [0, 1] \times UA$. Then X is diffeomorphic to $[0, 1] \times A \times S^2$ by a diffeomorphism induced by a parallelization $\tau$. The closed two-form $\omega$ is defined consistently on $\partial X$. It suffices to prove that the coboundary map $\partial$ of the long exact cohomology sequence associated to the pair $(X, \partial X)$ maps the class of $\omega|_{\partial X}$ to 0 in $H^3(X, \partial X)$. Since $H_3(X, \partial X)$ is Poincaré dual to

$$H_3(X) \cong (H_1(A) \otimes H_2(S^2)) \oplus (H_3(A) \otimes H_0(S^2)),$$

it is generated

* by classes of the form $[0, 1] \times s_+(\Sigma)$ for surfaces $\Sigma$ of A such that $\partial\Sigma \subset \partial A$ and for graphs $s_+(\Sigma)$ of sections in $UA$ associated to positive normals of the $\Sigma$, and

* by $[0, 1] \times \{a\} \times S^2$ for some $a \in A$, when $\partial A = \emptyset$.

The evaluation of $\partial[\omega|_{\partial X}]$ on these classes is the evaluation of $[\omega|_{\partial X}]$ on their boundary. It is clearly zero for $\partial[0, 1] \times \{a\} \times S^2$ since

$$\int_{\{(1,a)\} \times S^2} \omega = \int_{\{(0,a)\} \times S^2} \omega = 1.$$
Let us conclude the proof by proving

$$ \int_{\partial([0,1]\times s_+(\Sigma))} \omega = 0 $$

for a surface $\Sigma$ as above transverse to $\{-1\} \times \partial A$. This integral is invariant under the homotopies of $(\tau_0, \tau_1)$ that fix $(\tau_0, \tau_1)$ near $\partial A$. (See Lemma B.2.) Therefore, we can assume that $\tau_0 = \tau_1$ in a neighborhood $[-2,0] \times \partial A$ of $\partial A$ in $A$, that $\tau_0$ is a genuine parallelization in this neighborhood, and that the positive normal to $\Sigma$ is $\tau_0(\cdot \times e_3)$ along $\Sigma \cap (\{-1\} \times \partial A)$. Set $A_{-1} = A \setminus (]-1,0]\times \partial A)$ and $\Sigma_{-1} = \Sigma \cap A_{-1}$. Extend $\omega$ so that $\omega = p_{UA}^*(\omega(\tau_0))$ over $[0,1] \times UA|_{[-2,0]\times \partial A}$. We have $\int_{\partial([0,1]\times s_+(\Sigma))} \omega = \int_{\partial([0,1]\times s_+(\Sigma_{-1}))} \omega$. Proposition 19.22 ensures $\int_{\{0\}\times s_+(\Sigma_{-1})} \omega = \int_{\{1\}\times s_+(\Sigma_{-1})} \omega$. Since $\int_{[0,1]\times \partial s_+(\Sigma_{-1})} \omega = 0$, we get $\int_{\partial([0,1]\times s_+(\Sigma))} \omega = 0$. $\square$

**Proposition 19.24.** Let $A$ be a compact oriented 3-manifold equipped with three pseudo-parallelizations $\tau_0, \tau_1$, and $\tau_2$ that coincide with a common genuine parallelization along a regular neighborhood of $\partial A$. Let $n \in \mathbb{N}$. Let $\omega(\tau_0)$ and $\omega(\tau_1)$ be homogeneous boundary forms respectively associated with $(UA, \tau_0)$ and $(UA, \tau_1)$.

Under the assumptions of Proposition 19.23, as in Corollary 9.4, set

$$ z_n([0,1] \times UA; \omega) = \sum_{\Gamma \in \mathcal{D}_n^c} \zeta_{\Gamma} \int_{[0,1] \times \tilde{S}_{V(\Gamma)}(TA)} \bigwedge_{e \in E(\Gamma)} p_e^*(\omega) [\Gamma]. $$

If $A$ embeds in a rational homology 3-ball, then $z_n([0,1] \times UA; \omega)$ depends only on the pseudo-parallelizations $\tau_0$ and $\tau_1$. It is denoted by $z_n(A; \tau_0, \tau_1)$, and the following properties are satisfied.

* If $n$ is even, then $z_n(A; \tau_0, \tau_1) = 0$.

* If $B$ is a compact oriented 3-manifold embedded in the interior of $A$, if $\tau_0$ and $\tau_1$ coincide on a neighborhood of $A \setminus B$, and if $\tau_0$ restricts to a neighborhood of $\partial B$ as a genuine parallelization, then $z_n(B; \tau_0, \tau_1) = z_n(A; \tau_0, \tau_1)$.

* If $\tau_0$ and $\tau_1$ are actual parallelizations, then we have

$$ z_n(A; \tau_0, \tau_1) = \frac{p_1(\tau_0, \tau_1)}{4} \beta_n. $$

* We have $z_n(A; \tau_0, \tau_2) = z_n(A; \tau_0, \tau_1) + z_n(A; \tau_1, \tau_2)$. (In particular, we have $z_n(A; \tau_0, \tau_1) = -z_n(A; \tau_1, \tau_0)$.)
* For any orientation-preserving bundle isomorphism $\Psi$ of UA over the identity map of A, we have $z_n(A; \Psi \circ \tau_0, \Psi \circ \tau_1) = z_n(A; \tau_0, \tau_1)$.

* For any orientation-preserving diffeomorphism $\psi$ from A to another compact oriented 3-manifold B, we have

$$z_n(B; T\psi \circ \tau_0 \circ (\psi^{-1} \times 1_{\mathbb{R}^3}), T\psi \circ \tau_1 \circ (\psi^{-1} \times 1_{\mathbb{R}^3})) = z_n(A; \tau_0, \tau_1).$$

* If $\tau'_1$ is homotopic to $\tau_1$ relatively to $\partial A$ in the sense of Definition 19.21, then we have

$$z_n(A; \tau_0, \tau'_1) = z_n(A; \tau_0, \tau_1).$$

* For any orientation-preserving diffeomorphism $\psi_1$ of A isotopic to the identity map of A relatively to $\partial A$, we have

$$z_n(A; \tau_0, T\psi_1 \circ \tau_1 \circ (\psi_1^{-1} \times 1_{\mathbb{R}^3})) = z_n(A; \tau_0, \tau_1),$$

where $\psi_1$ is used to carry the required parametrization of $N(\gamma)$.

PROOF: In this proof, the manifold A embeds in a rational homology 3-ball. Lemma 9.12 implies that $z_n(A; \tau_0, \tau_1) = 0$ when $n$ is even. Assume that $n$ is odd from now on. Let us first prove that $z_n([0, 1] \times UA; \omega)$ does not depend on the closed extension $\omega$ when A is a rational homology 3-ball and when $\tau_0$ is standard near $\partial A$. According to Lemma 19.16, $\omega(\tau_0)$ (resp. $\omega(\tau_1)$) extends to a homogeneous propagating form of $(C_2(S^3(A/B_{S^3})), \tau_0)$ (resp. of $(C_2(S^3(A/B_{S^3})), \tau_1)$). Set $X = [0, 1] \times C_2(S^3(A/B_{S^3}))$. The above extensions of $\omega(\tau_0)$ and $\omega(\tau_1)$ together with $\omega$ (extended as $p_{\tau_0}^*(\omega_{S^2})$ on $[0, 1] \times (\partial C_2(S^3(A/B_{S^3})) \setminus UA))$ determine a closed 2-form of $\partial X$. This form extends as a closed form on $X$ by Lemma 9.1. Then Corollary 9.4 implies that $z_n([0, 1] \times UA; \omega)$ does not depend on $\omega$ when A is a rational homology 3-ball and when $\tau_0$ is standard near $\partial A$. In general, embed A in the interior of such a space $B_R$. The pseudo-parallelization $\tau_0$ extends to $B_R$ as a pseudo-parallelization standard near $\partial B_R$ according to Lemma 19.10. The form $\omega$ of Proposition 19.23 extends to $[0, 1] \times UB_R$ as $p_{UB_R}^*(\omega(\tau_0))$ on $[0, 1] \times U(B_R \setminus A)$. Then $z_n([0, 1] \times U(B_R \setminus A); \omega) = 0$ since $\bigwedge_{e \in E(\Gamma)} p_e^*(\omega)$ pulls back through $\check{S}_{V(\Gamma)}(T(B_R \setminus A))$ for any involved $\Gamma$. So $z_n([0, 1] \times UA; \omega) = z_n([0, 1] \times UB_R; \omega)$ is independent of $\omega|_{[0,1] \times UA}$.

Set $z_n(A; \omega(\tau_0), \omega(\tau_1)) = z_n([0, 1] \times UA; \omega)$. We easily observe

$$z_n(A; \omega(\tau_0), \omega(\tau_1)) = z_n(A; \omega(\tau_0), \omega(\tau_1)) + z_n(A; \omega(\tau_1), \omega(\tau_2)).$$

Assume $\tau_0 = \tau_1 = (N(\gamma); \tau_e, \tau_b)$. Assume that the forms $\omega(\tau_0) = \omega(\tau_0, F_0)$ and $\omega(\tau_1) = \omega(\tau_0, F_1)$ of Definition 19.13 are obtained from one another by
changing the map $F = F_0: [a, b] \times [-1, 1] \to SO(3)$ to another one $F_1$. There exists a homotopy $F_t$ from $F_0$ to $F_1$. Such a homotopy induces a homotopy

$$F_t(\gamma, \tau_b): \begin{array}{rcl} [a, b] \times \gamma \times [-1, 1] \times S^2 & \to & [a, b] \times \gamma \times [-1, 1] \times S^2 \\ (s, c, u; Y) & \mapsto & (s, c, u; F_t(s, u)(Y)). \end{array}$$

Then we have $z_n(A; \omega(\tau_0), \omega(\tau_1)) = z_n(N(\gamma); \omega(\tau_0, F_0), \omega(\tau_0, F_1))$. Use $\tau_b$ to identify $UN(\gamma)$ with $[a, b] \times \gamma \times [-1, 1] \times S^2$, and define $\omega(\gamma, \tau_b)$ on $[0, 1] \times [a, b] \times \gamma \times [-1, 1] \times S^2$ with respect to the formula for $\omega(\gamma, \tau_b)$ in Definition 19.13 by

$$\omega(\gamma, \tau_b) = \frac{p(\tau_b \circ \mathcal{T}_{\gamma}^{-1})^* (\omega_{S^2}) + p(\tau_b \circ F(\gamma, \tau_b)^{-1})^* (\omega_{S^2})}{2}.$$

This formula does not depend on the coordinate along $\gamma$. So $\omega$ pulls back through a projection from $[0, 1] \times UN(\gamma)$ to $[0, 1] \times [a, b] \times [-1, 1] \times S^2$. The $\bigwedge_{e \in E(\Gamma)} p_e^*(\omega)$ pull back through $[0, 1] \times \check{S}_{V(\Gamma)}(TN(\gamma)|_{[a,b]\times\{c\}\times[-1,1]})$. Hence $z_n([0, 1] \times UN(\gamma); \omega)$ vanishes. This proves that $z_n(N(\gamma); \omega(\tau_0, F_0), \omega(\tau_0, F_1))$ vanishes. We conclude that $z_n(A; \omega(\tau_0), \omega(\tau_1))$ depends only on $\tau_0$ and $\tau_1$.

For an orientation-preserving bundle isomorphism $\Psi$ of $UA$ over the identity map of $A$, the pseudo-parallelization

$$\Psi \circ (\tau_0 = (N(\gamma); \tau_e, \tau_b)) = (N(\gamma); \Psi \circ \tau_e, \Psi \circ \tau_b)$$

makes unambiguous sense. The following commutative diagram

<figure><img src="image_155.png" /></figure>

shows that $p(\tau_b) = p(\Psi \circ \tau_b) \circ \Psi$. So we have

$$\omega(\Psi \circ \tau_0, F) = (\Psi^{-1})^* \omega(\tau_0, F).$$

The form $\omega$ on $[0, 1] \times UA$ can be pulled back by the orientation-preserving $1_{[0,1]} \times \Psi^{-1}$, similarly. We get

$$z_n(A; \Psi \circ \tau_0, \Psi \circ \tau_1) = z_n(A; \tau_0, \tau_1).$$

Proposition 10.7 implies that $z_n(A; \tau_0, \tau_1) = \frac{p_1(\tau_0, \tau_1)}{4}\beta_n$ as soon as $\tau_0$ and $\tau_1$ are actual parallelizations and $A$ embeds in a rational homology ball to which $\tau_0$ extends as a genuine parallelization. Embed $A$ in a rational homology ball $B_R$. Let $\tau$ be a parallelization of $B_R$. Then $\tau|_A =$
$\Psi \circ \tau_0$ for an orientation-preserving bundle isomorphism $\Psi$ of UA. We have $z_n(A; \Psi \circ \tau_0, \Psi \circ \tau_1) = \frac{p_1(\Psi \circ \tau_0, \Psi \circ \tau_1)}{4} \beta_n$. The above behavior of $z_n$ under an orientation-preserving bundle isomorphism of UA allows us to conclude that $z_n(A; \tau_0, \tau_1) = \frac{p_1(\tau_0, \tau_1)}{4} \beta_n$ as soon as $\tau_0$ and $\tau_1$ are actual parallelizations.

For any orientation-preserving diffeomorphism $\psi$ from $A$ to $B$, the reader can check

$$z_n(B; T\psi \circ \tau_0 \circ (\psi^{-1} \times 1_{\mathbb{R}^3}), T\psi \circ \tau_1 \circ (\psi^{-1} \times 1_{\mathbb{R}^3})) = z_n(A; \tau_0, \tau_1)$$

as above.

If $\tau_1$ is homotopic to $\tau_0$ in the sense of Definition 19.21, then there exists a map $\Psi: [0, 1] \times UA \to UA$ such that $(t \mapsto \Psi(t, .) \circ \tau_0)$ is a homotopy of pseudo-parallelizations from $\tau_0$ to $\tau_1$. So we have

$$z_n(A; \tau_0, \tau_1) = z_n([0, 1] \times UA; \omega = (\Psi^{-1})^*(\omega(\tau_0, F))).$$

The form $\omega$ pulls back through a map from $[0, 1] \times UA$ to UA, and the forms $\bigwedge_{e \in E(\Gamma)} p_e^*(\omega)$ pull back through $\mathcal{S}_{V(\Gamma)}(TA)$. So $z_n([0, 1] \times UA; \omega)$ vanishes.

Let $\psi: [0, 1] \times A \to A$ be an isotopy, which maps $(t, u)$ to $\psi_t(u)$, such that $\psi_0 = 1$. It induces the homotopy

$$\Psi: \begin{array}{ccc} [0, 1] \times UA & \to & UA \\ (t, u) & \mapsto & \tau_1 \circ (\psi_t \times 1_{S^2}) \circ \tau_1^{-1} \circ T\psi_t^{-1}(u), \end{array}$$

which satisfies

$$\begin{aligned} p(T\psi_t \circ \tau_1 \circ (\psi_t^{-1} \times 1_{S^2})) &= p_{S^2} \circ \tau_1^{-1} \circ \tau_1 \circ (T\psi_t \circ \tau_1 \circ (\psi_t^{-1} \times 1_{S^2}))^{-1} \\ &= p(\tau_1) \circ \Psi(t, \cdot). \end{aligned}$$

We get

$$\begin{aligned} z_n(A; \tau_1 &= T\psi_0 \circ \tau_1 \circ (\psi_0^{-1} \times 1_{\mathbb{R}^3}), T\psi_1 \circ \tau_1 \circ (\psi_1^{-1} \times 1_{\mathbb{R}^3})) \\ &= z_n([0, 1] \times UA; (\Psi)^*(\omega(\tau_1))) = 0. \end{aligned}$$

□

The main result of this section is the following theorem.

**Theorem 19.25.** Let $A$ be a compact 3-manifold equipped with two pseudo-parallelizations $\tau_0$ and $\tau_1$ that coincide with a common genuine parallelization along a regular neighborhood of $\partial A$. Assume that $A$ embeds in a rational homology 3-ball. With the notation of Proposition 19.24, we have

$$z_n(A; \tau_0, \tau_1) = \frac{p_1(\tau_0, \tau_1)}{4} \beta_n$$

for any natural integer $n$.
To prove this theorem, we will prove it in special cases and show that these special cases are sufficient to get a complete proof.

**Lemma 19.26.** Assume $A = [2, 9] \times \gamma \times [-2, 2]$. Equip $A$ with a pseudo-parallelization $\tau_0 = (N(\tilde{\gamma}); \tau_e, \tau_b)$ such that

$$N(\tilde{\gamma}) = [3, 5] \times \gamma \times [-1, 1] \sqcup [6, 8] \times \gamma \times [-1, 1].$$

Then there exists a parallelization $\tau_1$ of $A$ that coincides with $\tau_e$ in a neighborhood of $\partial A$. Furthermore, we have

$$z_n(A; \tau_0, \tau_1) = \frac{p_1(\tau_0, \tau_1)}{4} \beta_n$$

for any such parallelization.

**PROOF:** We first prove the lemma for some chosen pseudo-parallelizations $\tilde{\tau}_0$ and $\tilde{\tau}_1$ satisfying the assumptions and behaving as “products by $\gamma$”. For these pseudo-parallelizations, this product behavior will imply $p_1(\tilde{\tau}_0, \tilde{\tau}_1) = 0$ and $z_n(A; \tilde{\tau}_0, \tilde{\tau}_1) = 0$. Define the parallelization

$$ \begin{array}{rcl} \tau_A: & A \times \mathbb{R}^3 & \to UA \\ & (s_0, c_0, u_0; e_1) & \mapsto \frac{d}{ds}(s, c_0, u_0)(s_0, c_0, u_0) \\ & (s_0, c_0, u_0; e_2) & \mapsto \frac{d}{dc}(s_0, c, u_0)(s_0, c_0, u_0) \\ & (s_0, c_0, u_0; e_3) & \mapsto \frac{d}{du}(s_0, c_0, u)(s_0, c_0, u_0) \end{array} $$

of $A$. Define $\tilde{\tau}_e: (A \setminus \mathring{N}(\tilde{\gamma})) \times \mathbb{R}^3 \to U(A \setminus \mathring{N}(\tilde{\gamma}))$ by

$$ \tilde{\tau}_e = \begin{cases} \tau_A & \text{on } A \setminus ([2, 8[ \times \gamma \times ]-1, 1[) \times \mathbb{R}^3 \\ \tau_A \circ \mathcal{T}_\gamma^{-1} & \text{on } [5, 6] \times \gamma \times ]-1, 1[) \\ \tau_A \circ \mathcal{T}_\gamma^{-2} & \text{on } [2, 3] \times \gamma \times ]-1, 1[), \end{cases} $$

where $\mathcal{T}_\gamma(t, c \in \gamma, u \in [-1, 1]; X \in \mathbb{R}^3) = (t, c, u; \rho_{\alpha(u)}(X))$ as in Definition 19.9. Define $\tilde{\tau}_b: N(\tilde{\gamma}) \times \mathbb{R}^3 \to UN(\tilde{\gamma})$ by

$$ \tilde{\tau}_b = \begin{cases} \tau_A & \text{on } [6, 8] \times \gamma \times ]-1, 1[ \\ \tau_A \circ \mathcal{T}_\gamma^{-1} & \text{on } [3, 5] \times \gamma \times ]-1, 1[. \end{cases} $$

Set $\tilde{\tau}_0 = (N(\tilde{\gamma}); \tilde{\tau}_e, \tilde{\tau}_b)$. Define a map

$$ \begin{array}{rcl} \tilde{F}: & [3, 8] \times [-1, 1] & \to SO(3) \\ & (3, u) & \mapsto \rho_{-2\alpha(u)} \\ & (8, u) & \mapsto 1_{SO(3)} \\ & (t, \pm 1) & \mapsto 1_{SO(3)}. \end{array} $$
Finally define $\tilde{\tau}_1$ such that

$$ \tilde{\tau}_1 = \begin{cases} \tau_A & \text{on } A \setminus ([2, 8[ \times \gamma \times ]-1, 1]) \times \mathbb{R}^3 \\ \tau_A \circ \mathcal{T}_\gamma^{-2} & \text{on } [2, 3] \times \gamma \times ]-1, 1] \end{cases} $$

and

$$ \tilde{\tau}_1(s, c, u; X) = (\tau_A(s, c, u; \tilde{F}(s, u)(X))) $$

when $(s, u) \in [3, 8] \times [-1, 1]$.

Let us prove $p_1(\tilde{\tau}_0, \tilde{\tau}_1) = 0$. The involved trivializations of $T([0, 1] \times A) \otimes C$ on $\partial([0, 1] \times A)$ are obtained from the natural parallelization $T([0, 1] \oplus \tau_A)$ by composition by a map from $\partial([0, 1] \times A) = \gamma \times \partial([0, 1] \times [2, 9] \times [-2, 2])$ to $SU(4)$, which does not depend on the coordinate along $\gamma$. Since $\pi_2(SU(4)) = \{0\}$, this map extends to $SU(4)$.

Let us similarly prove $z_n(A; \tilde{\tau}_0, \tilde{\tau}_1) = 0$. Set $Y = [0, 1] \times S^2 \times [2, 9] \times [-2, 2]$. The parallelization $\tau_A$ identifies $[0, 1] \times UA$ with $\gamma \times Y$. We have $z_n(A; \tilde{\tau}_0, \tilde{\tau}_1) = z_n([0, 1] \times UA; \omega)$ for a closed two-form $\omega$ whose restriction $\omega|_{\partial}$ to $\partial([0, 1] \times UA)$ factors through the projection of $\gamma \times \partial Y$ onto $\partial Y$. The involved closed 2-form on $\partial Y$ extends to $Y$ as a closed form $\omega_Y$ since $\omega|_{\partial}$ extends to the whole $[0, 1] \times UA$, according to Proposition 19.23. Then $\omega$ can be chosen as the pull-back of $\omega_X$ under the projection of $\gamma \times X$ onto $X$. Thus, the forms $\bigwedge_{e \in E(\Gamma)} p_e^*(\omega)$ again pull back through a projection onto a space of dimension smaller than the degree of the forms. So $z_n([0, 1] \times UA; \omega)$ vanishes.

There exists an orientation-preserving bundle isomorphism $\Psi$ of $UA$ over the identity map of $A$, such that $\tau_0 = \Psi \circ \tilde{\tau}_0$. The parallelization $\tau_1 = \Psi \circ \tilde{\tau}_1$ satisfies the assumptions of the lemma. We have $z_n(A; \Psi \circ \tilde{\tau}_0, \Psi \circ \tilde{\tau}_1) = z_n(A; \tilde{\tau}_0, \tilde{\tau}_1) = 0$ and $p_1(\Psi \circ \tilde{\tau}_0, \Psi \circ \tilde{\tau}_1) = p_1(\tilde{\tau}_0, \tilde{\tau}_1) = 0$. So we have $z_n(A; \tau_0, \tau_1) = \frac{p_1(\tau_0, \tau_1)}{4} \beta_n$. We conclude for any another parallelization $\tau'_1$ that coincides with $\tau_e$ near $\partial A$, because $z_n(A; \tau_1, \tau'_1) = \frac{p_1(\tau_1, \tau'_1)}{4} \beta_n$. $\square$

**Lemma 19.27.** Let $A$ be a compact oriented 3-manifold that embeds in a rational homology 3-ball. Let $[-7, 0] \times \partial A$ be a collar neighborhood of $A$. Let $\gamma \times [-2, 2]$ be a disjoint union of annuli in $\partial A$. Set $N(\gamma) = [-2, -1] \times \gamma \times [-1, 1]$. Let $\tau_0 = (N(\gamma); \tau_e, \tau_b)$ be a pseudo-parallelization of $A$ that coincides with the restriction of a parallelization $\tau_1$ of $A$ in a neighborhood of $\partial A$. Then we have

$$ z_n(A; \tau_0, \tau_1) = \frac{p_1(\tau_0, \tau_1)}{4} \beta_n. $$

**PROOF:** Recall $A_{-2} = A \setminus ([]-2, 0] \times \partial A)$. Figure 19.1 shows the schema of the proof.
<figure><img src="image_156.png" /><figcaption>Figure 19.1: Schema of proof for Lemma 19.27</figcaption></figure>

Let $f: [-7, 0] \to [-7, -2]$ be a diffeomorphism such that $f(t) = t-2$ when $t \ge -3$, and $f(t) = t$ when $t \le -6$. Let $\psi: A \to A_{-2}$ be a diffeomorphism restricting to $A \setminus (]-7, 0] \times \partial A)$ as the identity map, and mapping $(t, x) \in [-7, 0] \times \partial A$ to $(f(t), x)$.

There exists a bundle isomorphism $\Phi$ of $UA_{-2}$ over the identity map of $A_{-2}$ such that $\tau_e|_{A_{-2}} = \Phi \circ T\psi \circ \tau_1 \circ (\psi^{-1} \times 1_{\mathbb{R}^3})$. Let $\tau_{-1}$ be the pseudo-parallelization of $A$ that coincides with $\tau_0$ over $[-2, 0] \times \Sigma$, and with $\Phi \circ T\psi \circ \tau_0 \circ (\psi^{-1} \times 1_{\mathbb{R}^3})$ over $A_{-2}$. Then $\tau_{-1}$ is a parallelization outside $[-7, 0] \times \gamma \times [-2, 2]$. Since $[-7, 0] \times \gamma \times [-2, 2]$ satisfies the hypotheses of Lemma 19.26 up to reparametrization, Lemma 19.26 and Proposition 19.24 ensure that

$$z_n(A; \tau_{-1}, \tau_1) = \frac{p_1(\tau_{-1}, \tau_1)}{4} \beta_n.$$

To conclude, we prove that $z_n(A; \tau_0, \tau_1) = \frac{1}{2}z_n(A; \tau_{-1}, \tau_1)$ and $p_1(\tau_0, \tau_1) = \frac{1}{2}p_1(\tau_{-1}, \tau_1)$. The element $z_n(A; \tau_{-1}, \tau_0)$ of $\mathcal{A}_n(\emptyset)$ can be written as

$$z_n(A_{-2}; \Phi \circ T\psi \circ \tau_0 \circ (\psi^{-1} \times 1_{\mathbb{R}^3}), \Phi \circ T\psi \circ \tau_1 \circ (\psi^{-1} \times 1_{\mathbb{R}^3})) = z_n(A; \tau_0, \tau_1).$$
We similarly have $p_1(\tau_{-1}, \tau_0) = p_1(\tau_0, \tau_1)$. $\square$

**Lemma 19.28.** Let $\Sigma$ be a compact oriented surface. Let $\gamma_0$ and $\gamma_1$ be two disjoint unions of curves of $\Sigma$ with respective tubular neighborhoods $\gamma_0 \times [-1, 1]$ and $\gamma_1 \times [-1, 1]$. Set $A = [0, 3] \times \Sigma$, $N(\gamma_0) = [1, 2] \times \gamma_0 \times [-1, 1]$ and $N(\gamma_1) = [1, 2] \times \gamma_1 \times [-1, 1]$.

There exist two pseudo-parallelizations $\tau_0 = (N(\gamma_0); \tau_{e,0}, \tau_{b,0})$ and $\tau_1 = (N(\gamma_1); \tau_{e,1}, \tau_{b,1})$ that coincide near $\partial A$ if and only if $\gamma_0$ and $\gamma_1$ have the same class in $H_1(\Sigma; \mathbb{Z}/2\mathbb{Z})$. In this case, we have

$$z_n(A; \tau_0, \tau_1) = \frac{p_1(\tau_0, \tau_1)}{4} \beta_n.$$
PROOF: Without loss of generality, assume that $\Sigma$ is connected and that the complement of $(\gamma_0 \times [-1, 1]) \cup (\gamma_1 \times [-1, 1])$ in $\Sigma$ is not empty. Let $\tau$ be a parallelization of $A = [0, 3] \times \Sigma$.

Assume that $\tau_0$ and $\tau_1$ are two pseudo-parallelizations as in the statement, which coincide near $\partial A$. Let us prove that $\gamma_0$ and $\gamma_1$ are homologous modulo $2H_1(\Sigma; \mathbb{Z})$.

If the boundary of $\Sigma$ is empty, choose a disk $D$ of $\Sigma$ outside $\gamma_0 \times [-1, 1] \cup \gamma_1 \times [-1, 1]$ and assume that $\tau_{e,0}$ and $\tau_{e,1}$ coincide on $[0, 3] \times D$, without loss of generality. This allows us to assume $\partial\Sigma \neq \emptyset$, without loss of generality, by possibly removing the interior of $D$ from $\Sigma$. For $i \in \{0, 1\}$ and for the pseudo-parallelization $\tau_i$ of the statement, write $\tau_{e,i}$ as $\tau \circ \psi_{\mathbb{R}}(g_i)$ for some $g_i: A \setminus \tilde{N}(\gamma_i) \to SO(3)$, with the notation of Section 4.2. Then the restriction of $g_i$ to a meridian curve of $\gamma_i$ is not homotopic to a constant loop. The maps $g_0$ and $g_1$ coincide near $\partial A$. Let $c: [0, 1] \to \Sigma$ be a path such that $c(0)$ and $c(1)$ are in $\partial\Sigma$. Then the restriction of $g_i$ to

$$ (\{3\} \times c([0, 1])) \cup (-[0, 3] \times c(1)) \cup (-\{0\} \times c([0, 1])) \cup ([0, 3] \times c(0)) $$

is null-homotopic if and only if the mod 2 intersection of $c$ with $\gamma_i$ is trivial. So $\gamma_0$ and $\gamma_1$ must be homologous modulo $2H_1(\partial A; \mathbb{Z})$.

Conversely, assume that $\gamma_0$ and $\gamma_1$ are homologous modulo $2H_1(\Sigma; \mathbb{Z})$. Define $g_0: A \setminus \tilde{N}(\gamma_0) \to SO(3)$ to be the map sending $A \setminus ([0, 2] \times \gamma_0 \times [-1, 1])$ to the identity of $SO(3)$, and mapping $(t, c, u) \in ([0, 1] \times \gamma_0 \times [-1, 1])$ to $\rho_{-\alpha(u)}$. First define $g_1: A \setminus \tilde{N}(\gamma_1) \to SO(3)$ on $(A \setminus ([0, 1[ \times \Sigma)) \setminus \tilde{N}(\gamma_1)$ so that $g_1$ sends $(A \setminus ([0, 1] \times \Sigma)) \setminus \tilde{N}(\gamma_1)$ to the identity of $SO(3)$ and $g_1$ maps $(t, c, u) \in (\{1\} \times \gamma_1 \times [-1, 1])$ to $\rho_{-\alpha(u)}$. Since the classes of $\gamma_0$ and $\gamma_1$ coincide in $H_1(\Sigma; \mathbb{Z}/2\mathbb{Z})$, the restrictions to $\{1\} \times \Sigma$ of $g_1$ and $g_0$ are homotopic on the one-skeleton of $\Sigma$, and hence on $\Sigma$. This allows us to extend $g_1$ to $[0, 1] \times \Sigma$ so that $g_0$ and $g_1$ coincide in a neighborhood of $\partial A$. Then, for $i \in \{0, 1\}$, there exists $\tau_{b,i}$ such that $\tau_i = (N(\gamma_i); \tau_{e,i} = \tau \circ \psi_{\mathbb{R}}(g_i), \tau_{b,i})$ is a pseudo-parallelization. The pseudo-parallelizations $\tau_0$ and $\tau_1$ coincide near $\partial A$.

Set $B = [0, 6] \times \Sigma$ and $N(\gamma'_0) = [4, 5] \times \gamma_0 \times [-1, 1]$. Extend $\tau_0$ to $B$ as a pseudo-parallelization $\tau_{0,B} = (N(\gamma_0) \cup N(\gamma'_0); \tau_{e,B,0}, \tau_{b,B,0})$. Extend $\tau_1$ to $B$ as a pseudo-parallelization $\tau_{1,B}$ that coincides with $\tau_{0,B}$ on $[3, 6] \times \Sigma$. We have $z_n(A; \tau_0, \tau_1) = z_n(B; \tau_{0,B}, \tau_{1,B})$ and $p_1(\tau_0, \tau_1) = p_1(\tau_{0,B}, \tau_{1,B})$.

According to Lemma 19.26, there is a parallelization $\tau_2$ of $B$ that coincides with $\tau_{0,B}$ near $\partial B$, and we have $z_n(B; \tau_{0,B}, \tau_2) = \frac{p_1(\tau_{0,B}, \tau_2)}{4} \beta_n$.

Apply Lemma 19.27 to prove that $z_n(B; \tau_{1,B}, \tau_2) = \frac{p_1(\tau_{1,B}, \tau_2)}{4} \beta_n$. (To apply Lemma 19.27 as it is stated in $B$, first rotate $N(\gamma_0) = [1, 2] \times \gamma_0 \times [-1, 1]$
around $\gamma_0$ by an isotopy that sends $1 \times [-1, 1]$ to $2 \times [1, -1]$ and apply Proposition 19.24.) This implies that $z_n(B; \tau_{0,B}, \tau_{1,B}) = \frac{p_1(\tau_{0,B}, \tau_{1,B})}{4}$. $\square$

**Lemma 19.29.** Let $\Sigma$ be a compact connected oriented surface with boundary. Let $\gamma, \gamma_0$, and $\gamma_1$ be three disjoint unions of curves of $\Sigma$ with respective tubular neighborhoods $\gamma \times [-1, 1]$, $\gamma_0 \times [-1, 1]$, and $\gamma_1 \times [-1, 1]$.<fnref n="1" /> Assume that $[\gamma_1] = [\gamma_0] + [\gamma]$ in $H_1(\Sigma; \mathbb{Z}/2\mathbb{Z})$. Set $A = [0, 6] \times \Sigma$, $N(\gamma) = [4, 5] \times \gamma \times [-1, 1]$, $N(\gamma_0) = [1, 2] \times \gamma_0 \times [-1, 1]$, and $N(\gamma_1) = [1, 2] \times \gamma_1 \times [-1, 1]$. Let $\tau_0 = (N(\gamma) \sqcup N(\gamma_0); \tau_{e,0}, \tau_{b,0})$ and $\tau_1 = (N(\gamma_1); \tau_{e,1}, \tau_{b,1})$ be two pseudo-parallelizations which coincide near $\partial A$. Then we have

$$z_n(A; \tau_0, \tau_1) = \frac{p_1(\tau_0, \tau_1)}{4} \beta_n.$$

**PROOF:** Lemma 19.28 allows us to choose arbitrary representatives of $[\gamma]$ and $[\gamma_0]$ for the proof, without loss of generality. In particular, there is no loss of generality in assuming that $\gamma_0$ is connected and that the intersection of $\gamma$ and $\gamma_0$ has no more than one point.

If $\gamma$ and $\gamma_0$ are disjoint, then we can perform an isotopy in $A$ to lower $\gamma$, and the result is a direct consequence of Lemma 19.28.

Assume that the intersection of $\gamma$ and $\gamma_0$ has one transverse point. Attach two copies $\Sigma$ and $\Sigma'$ of $\Sigma$ to a disk $D$ along intervals $I$ and $I'$. Let $\gamma'$, $\gamma'_0$, and $\gamma'_1$ be the respective copies of $\gamma$, $\gamma_0$, and $\gamma_1$ in $\Sigma'$. Let $\tilde{\Sigma} = \Sigma \cup D \cup \Sigma'$ as in Figure 19.2.

<figure><img src="image_157.png" /><figcaption>Figure 19.2: $\tilde{\Sigma}$</figcaption></figure>

Let $B = [0, 6] \times \tilde{\Sigma}$. Let $\tau_{B,0}$ be a pseudo-parallelization of $B$ extending the pseudo-parallelization $\tau_0$ used both for $A$ and for $A' = [0, 6] \times \Sigma'$. Let $\tau_{B,1}$ be a pseudo-parallelization that coincides with $\tau_1$ on $A$ and on $A'$, and with $\tau_{B,0}$ on $[0, 6] \times D$. Then we have $z_n(A; \tau_0, \tau_1) = \frac{1}{2}z_n(B; \tau_{B,0}, \tau_{B,1})$ and $p_1(\tau_0, \tau_1) = \frac{1}{2}p_1(\tau_{B,0}, \tau_{B,1})$. Since the intersection of $(\gamma \cup \gamma')$ and $(\gamma_0 \cup \gamma'_0)$ is zero modulo $2H_1(\partial A; \mathbb{Z})$, the homology classes of these curves can be represented by curves that do not intersect. So we have $z_n(B; \tau_{B,0}, \tau_{B,1}) = \frac{p_1(\tau_{B,0}, \tau_{B,1})}{4} \beta_n$. $\square$

**PROOF OF THEOREM 19.25:** Let us first prove the theorem when $A$ is a rational homology ball, according to the schema of Figure 19.3. Then there ex-

<footnote n="1">¹The disjoint union of curves $\gamma$ may intersect $\gamma_0$ and $\gamma_1$, and $\gamma_0$ may intersect $\gamma_1$.</footnote>
ists a parallelization $\tau_2$ of A that coincides with $\tau_0 = (N(\gamma); \tau_e, \tau_b)$ in a neighborhood of $\partial A$. Thicken the neighborhood $N(\gamma)$ to $[a-7, b+7] \times \gamma \times [-2, 2]$. Add bands to $\gamma \times [-2, 2]$ so that the disjoint union $\gamma \times [-2, 2]$ is embedded in a connected oriented surface $\Sigma$ of A with one boundary component. Let $[a-7, b+7] \times \Sigma$ be embedded in A so that this parametrization matches the previous one.

<figure><img src="image_158.png" /><figcaption>Figure 19.3: Schema of proof for Theorem 19.25</figcaption></figure>

After a possible homotopy of $\tau_2$, there exist annuli $\gamma_a \times [-1, 1]$ and $\gamma_b \times [-1, 1]$ in $\Sigma$ such that

* $\tau_0$ coincides with $\tau_2$ in a neighborhood of $[a-7, b+7] \times \partial\Sigma$,

* $\tau_0|_{\{a-4\} \times \Sigma} = \tau_2 \circ \mathcal{T}_{\gamma_a}$, and

* $\tau_0|_{\{b+4\} \times \Sigma} = \tau_2 \circ \mathcal{T}_{\gamma_b}^{-1}$.

Set

$$ N(\gamma_3) = N(\gamma) \sqcup ([b+2, b+3] \sqcup [b+5, b+6]) \times \gamma_b \times [-1, 1] \\ \sqcup ([a-6, a-5] \sqcup [a-3, a-2]) \times \gamma_a \times [-1, 1]. $$

Let $\tau_3 = (N(\gamma_3); \tau_{3,e}, \tau_{3,b})$ be a pseudo-parallelization, which coincides with $\tau_0$ outside $([b+1, b+7]) \times \gamma_b \times [-1, 1] \sqcup ([a-7, a-1]) \times \gamma_a \times [-1, 1]$, and which coincides with $\tau_2$ on $\{a-4\} \times \Sigma$ and on $\{b+4\} \times \Sigma$. According to Lemma 19.26 and to Proposition 19.24, we have

$$ z_n(A; \tau_0, \tau_3) = \frac{p_1(\tau_0, \tau_3)}{4} \beta_n, $$
as the left part of Figure 19.3 shows.

Set $B = [a-4, b+4] \times \Sigma$ and $C = A \setminus (]a-4, b+4[ \times \mathring{\Sigma})$. We have

$$z_n(A; \tau_2, \tau_3) = z_n(B; \tau_2, \tau_3) + z_n(C; \tau_2, \tau_3),$$

and $p_1$ decomposes similarly.

Let us now prove

$$z_n(C; \tau_2, \tau_3) = \frac{1}{4} p_1(\tau_2|_C, \tau_3|_C) \beta_n.$$

To do this, we apply Lemma 19.27, after an isotopy of $[b+4, b+7] \times \Sigma$ which sends $[b+5, b+6] \times \gamma_b \times [-1, 1]$ to itself (at the end) so that $\{b+5\} \times \gamma_b \times [-1, 1]$ is sent to $\{b+6\} \times \gamma_b \times (-[-1, 1])$, and $\{b+6\} \times \gamma_b \times [-1, 1]$ is sent to $\{b+5\} \times \gamma_b \times (-[-1, 1])$.

Let $\tau_4 = (([a-3, a-2] \sqcup [b+2, b+3]) \times \gamma_a \times [-1, 1]; \tau_{4,e}, \tau_{4,b})$ be a pseudo-parallelization of $B$ that coincides with $\tau_2$ in a neighborhood of $\partial B$. Lemma 19.26 implies

$$z_n(B; \tau_4, \tau_2) = \frac{1}{4} p_1(\tau_4, \tau_2|_B) \beta_n.$$

Since $[\gamma_a] + [\gamma] + [\gamma_b] = 0$ in $H_1(\Sigma; \mathbb{Z}/2\mathbb{Z})$, Lemma 19.29 and Proposition 19.24 imply $z_n(B; \tau_3, \tau_4) = \frac{1}{4} p_1(\tau_3|_B, \tau_4) \beta_n$. We get $z_n(B; \tau_2, \tau_3) = \frac{1}{4} p_1(\tau_2|_B, \tau_3|_B) \beta_n$, and hence

$$z_n(A; \tau_2, \tau_3) = \frac{1}{4} p_1(\tau_2, \tau_3) \beta_n.$$

So we have $z_n(A; \tau_0, \tau_2) = \frac{1}{4} p_1(\tau_0, \tau_2) \beta_n$. For the same reasons, we have $z_n(A; \tau_1, \tau_2) = \frac{1}{4} p_1(\tau_1, \tau_2) \beta_n$. Hence the lemma is proved when $A$ is a rational homology ball.

In general, the manifold $A$ is assumed to embed into a rational homology ball $B$, the pseudo-parallelization $\tau_0$ on $A$ extends to a pseudo-parallelization $\tilde{\tau}_0$ of $B$, and the pseudo-parallelization $\tau_1$ over $A$ extends to a pseudo-parallelization $\tilde{\tau}_1$ of $B$, which coincides with $\tilde{\tau}_0$ over $B \setminus \mathring{A}$. We have

$$z_n(A; \tau_0, \tau_1) = z_n(B; \tilde{\tau}_0, \tilde{\tau}_1) = \frac{p_1(\tilde{\tau}_0, \tilde{\tau}_1)}{4} \beta_n = \frac{p_1(\tau_0, \tau_1)}{4} \beta_n.$$

□
## 19.5 Proof of Theorem 19.17

**Proposition 19.30.** Let $\tilde{R}$ be a rational homology $\mathbb{R}^3$ equipped with an asymptotically standard pseudo-parallelization $\tau$. Let $\omega(\tau)$ be a homogeneous propagating form of $(C_2(R), \tau)$. Let $n \in \mathbb{N}$. With the notation of Corollary 10.9 and Notation 7.16, we have

$$z_n(\tilde{R}, \omega(\tau)) = \mathfrak{z}_n(R) + \frac{1}{4}p_1(\tau)\beta_n$$

and

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

**PROOF:** Let $\tau_0$ and $\tau_1$ be two pseudo-parallelizations of $\tilde{R}$ standard outside $B_R$. Let $\omega$ be a 2-form on $[0, 1] \times UB_R$ as in Lemma 9.1. Corollary 9.4 implies

$$z_n(\tilde{R}, \omega(\tau_1)) - z_n(\tilde{R}, \omega(\tau_0)) = z_n([0, 1] \times UB_R; \omega),$$

while Theorem 19.25 implies

$$z_n([0, 1] \times UB_R; \omega) = \frac{1}{4}(p_1(\tau_1) - p_1(\tau_0))\beta_n.$$

Conclude with Corollary 10.9. □

**PROOF OF THEOREM 19.17:** Theorem 12.7 implies

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

for any actual parallelization $\tau$ of $C$. We want to prove the same equality when $\tau$ is a pseudo-parallelization that is an actual parallelization over a tubular neighborhood $N(L)$ of $L$. Recall Lemma 7.27. Proposition 19.30 leaves us with the proof that

$$\tilde{Z}(C, L, \tau') = \prod_{j=1}^{k} \left( \exp(I_{\theta}(K_j, \tau')\alpha) \#_j \right) \tilde{Z}(C, L)$$

for any pseudo-parallelization $\tau'$ of $C$ that is an actual parallelization over a tubular neighborhood $N(L)$ of $L$.

First assume that the restriction of $\tau'$ to the tubular neighborhood $N(L)$ of $L$ extends to an actual parallelization $\tau$ of $C$. In this case, we apply
Proposition 14.49 with a closed 2-form $\tilde{\omega}$ on $[0, 1] \times UC$ as in its statement. Thus, we get

$$ \check{Z}(C, L, \tau') = \left( \prod_{j=1}^{k} \exp(I_j) \#_j \right) \check{Z}(C, L, \tau), $$

where $I_j$ is defined from $\tilde{\omega}$ for any component $K_j$ of $L = \bigsqcup_{j=1}^k K_j$ in Proposition 14.49. Let $p_{UN(L)}: [0, 1] \times UN(L) \to UN(L)$ denote the projection to the second factor. According to Proposition 19.23, the form $\tilde{\omega}$ may be expressed as $p_{UN(L)}^*(\omega_{S^2})$ over $[0, 1] \times UN(L)$. This factorization implies that the $I_j$ vanish. So, we have $\check{Z}(C, L, \tau') = \check{Z}(C, L, \tau)$. The degree one part of this equality implies $I_\theta(K_j, \tau') = I_\theta(K_j, \tau)$. Hence the theorem is proved when the restriction of $\tau'$ to the tubular neighborhood $N(L)$ of $L$ extends to an actual parallelization $\tau$ of $C$.

Recall that $D_r$ is the disk of the complex numbers of module less than or equal to $r$. Let $\tau_s$ denote both the standard parallelization of $\mathbb{R}^3$ and its restriction to $D_4 \times [-2, 2]$. Consider the neighborhood

$$ N(\gamma_2) = (D_3 \setminus \check{D}_1) \times [-1, 1] $$

of $(\gamma_2 = \partial D_2 \times \{0\})$. Let $\tau_2 = (N(\gamma_2); \tau_e, \tau_b)$ be a pseudo-parallelization of $D_4 \times [-2, 2]$ that coincides with $\tau_s$ in a neighborhood of $\partial(D_4 \times [-2, 2])$ and that maps $e_3$ to the vertical direction of $\{0\} \times [-2, 2]$ along $\{0\} \times [-2, 2]$.

Let $\tau_s^C$ and $\tau_2^C$ be two pseudo-parallelizations of $C$ that satisfy the following set $(*)$ $(\tau_s, \tau_2, C, L, K_j)$ of assumptions: They are actual parallelizations over a tubular neighborhood $N(L)$ of $L$. There is an embedding of $D_4 \times [-2, 2]$ in the rational homology cylinder $C$ equipped with the long tangle representative $L$ so that (the image of) $D_4 \times [-2, 2]$ intersects (the image of) $L$ along $\{0\} \times [-2, 2]$ and the orientations of $\{0\} \times [-2, 2]$ and $L$ match. With respect to this embedding, $\tau_s^C$ and $\tau_2^C$ respectively coincide with $\tau_s$ and $\tau_2$ on the image of $D_4 \times [-2, 2]$, and they coincide with each other elsewhere. The component of $L$ that intersects $D_4 \times [-2, 2]$ is denoted by $K_j$.

According to Proposition 19.23, there exists a closed 2-form $\tilde{\omega}$ on $[0, 1] \times U(D_4 \times [-2, 2])$ that restricts to

$$ (\{0\} \times U(D_4 \times [-2, 2])) \cup ([0, 1] \times U(D_4 \times [-2, 2])|_{\partial(D_4 \times [-2, 2])}) $$

as $p_{\tau_s}^*(\omega_{S^2})$, and to $\{1\} \times U(D_4 \times [-2, 2])$ as a homogeneous propagating form of $C_2(R(C), \tau_2^C)$ does. This closed 2-form is actually independent of $(C, L)$, so is the induced quantity $I(\tau_2)$ of Proposition 14.49 such that

$$ \check{Z}(C, L, \tau_2^C) = \exp(I(\tau_2)) \#_j \check{Z}(C, L, \tau_s^C). $$
In particular, the difference $(I_{\theta}(K_j, \tau_2^C) - I_{\theta}(K_j, \tau_s^C))$ is a constant $\ell(\tau_2)$. It can be obtained from the degree one part of $I(\tau_2)$.

Apply this computation when $R(C)$ is $SO(3)$, ($L = K$) is a knot, the homology class of $K$ represents the generator of $H_1(SO(3); \mathbb{Z}) = \mathbb{Z}/2\mathbb{Z}$, and $\tau_s^C$ is an actual parallelization. The two homotopy classes of parallelizations of $N(K)$ are obtained from one another by composition by the restriction to $N(K) \times \mathbb{R}^3$ of the map

$$ \begin{array}{ccc} SO(3) \times \mathbb{R}^3 & \to & SO(3) \times \mathbb{R}^3 \\ (\rho, x) & \mapsto & (\rho, \rho(x)). \end{array} $$

So, all parallelizations of $N(K)$ extend to $C$. In particular, $\tau_2^C|_{N(K)}$ extends to $C$ as a parallelization standard near $\partial C$.

Then the first studied case implies $I(\tau_2) = (I_{\theta}(K, \tau_2^C) - I_{\theta}(K, \tau_s^C)) \alpha = \ell(\tau_2)\alpha$. So we get $I(\tau_2) = (I_{\theta}(K_j, \tau_2^C) - I_{\theta}(K_j, \tau_s^C)) \alpha$, and hence

$$ \prod_{\ell=1}^{k} \left( \exp(-I_{\theta}(K_{\ell}, \tau_{2}^{C}) \alpha) \#_{\ell} \right) \check{Z}(C, L, \tau_{2}^{C}) = \prod_{\ell=1}^{k} \left( \exp(-I_{\theta}(K_{\ell}, \tau_{s}^{C}) \alpha) \#_{\ell} \right) \check{Z}(C, L, \tau_{s}^{C}) $$

for any two pseudo-parallelizations $\tau_s^C$ and $\tau_2^C$ that satisfy $\alpha(\tau_s, \tau_2, C, L, K_j) = \alpha(\tau_2, \tau_s, C, L, K_j)$.

Let $\tau'$ be a pseudo-parallelization of $C$ that coincides with an actual parallelization $\tau_N$ of $C$ on $N(L)$. Let $\tau$ be a parallelization of $C$. The restrictions of $\tau_N$ and $\tau$ to $\partial N(L)$ are homotopic along the meridians of $L$. They differ by the action of the generator of $\pi_1(SO(2))$ along parallels on components $K_j$ for $K_j$ in some finite set $A$. If $A = \emptyset$, the first studied case implies

$$ \prod_{j=1}^{k} \left( \exp(-I_{\theta}(K_j, \tau')\alpha) \#_j \right) \check{Z}(C, L, \tau') = \check{Z}(C, L). $$

Otherwise, equip each component $K_j$ of $A$ with one embedding of $D_4 \times [-2, 2]$ whose image meets $L$ in $K_j$ along $\{0\} \times [-2, 2]$, so that the orientations of $\{0\} \times [-2, 2]$ and $L$ match. Perform a homotopy of $\tau'$ as in Definition 19.21 to transform $\tau'$ to a pseudo-parallelization $\tau''$ such that $\tau''$ is induced by $\tau_s$ on the image of the above embeddings. For $B \subseteq A$, let $\tau_B''$ be obtained from $\tau''$ by changing $\tau_s$ to $\tau_2$ on the images of the embeddings of $D_4 \times [-2, 2]$ that meet an element of $B$. Then for $B \subseteq A$ and for $K_{\ell} \in A \setminus B$, the pseudo-parallelizations $\tau_B''$ and $\tau_{B \cup \{K_{\ell}\}}''$ satisfy $\alpha(\tau_B'', \tau_{B \cup \{K_{\ell}\}}'') = \alpha(\tau_B'', \tau_{B \cup \{K_{\ell}\}}'')$.
we have

$$ \prod_{j=1}^{k} (\exp(-I_{\theta}(K_j, \tau''_{B \cup \{K_{\ell}\}})\alpha)_{\#j}) \check{Z}(\mathcal{C}, L, \tau''_{B \cup \{K_{\ell}\}}) \\ = \prod_{j=1}^{k} (\exp(-I_{\theta}(K_j, \tau''_{B})\alpha)_{\#j}) \check{Z}(\mathcal{C}, L, \tau''_{B}). $$

We get

$$ \prod_{j=1}^{k} (\exp(-I_{\theta}(K_j, \tau''_{A})\alpha)_{\#j}) \check{Z}(\mathcal{C}, L, \tau''_{A}) = \prod_{j=1}^{k} (\exp(-I_{\theta}(K_j, \tau'')\alpha)_{\#j}) \check{Z}(\mathcal{C}, L, \tau'') $$

by induction on $|A|$. Since $\tau''_A$ and $\tau$ are homotopic on $N(L)$, the first studied case implies

$$ \prod_{j=1}^{k} (\exp(-I_{\theta}(K_j, \tau''_{A})\alpha)_{\#j}) \check{Z}(\mathcal{C}, L, \tau''_{A}) = \check{Z}(\mathcal{C}, L). $$

So it suffices to prove that

$$ \prod_{j=1}^{k} (\exp(-I_{\theta}(K_j, \tau'')\alpha)_{\#j}) \check{Z}(\mathcal{C}, L, \tau'') = \prod_{j=1}^{k} (\exp(-I_{\theta}(K_j, \tau')\alpha)_{\#j}) \check{Z}(\mathcal{C}, L, \tau') $$

or that $\prod_{j=1}^{k} (\exp(-I_{\theta}(K_j, \tau')\alpha)_{\#j}) \check{Z}(\mathcal{C}, L, \tau')$ is invariant by a homotopy of $\tau'$, as in Definition 19.21, supported in a ball where $\tau'$ is a genuine parallelization (namely, around an image of $D_4 \times [-2, 2]$).

Again, the effect on $\check{Z}(\mathcal{C}, L, \tau')$ of such a homotopy depends only on the homotopy inside the ball, according to Propositions 14.49 and 19.23. Since such a ball equipped with the homotopy may be inserted in a tangle equipped with a genuine trivialization $\tau$, we conclude that

$$ \prod_{j=1}^{k} (\exp(-I_{\theta}(K_j, \tau')\alpha)_{\#j}) \check{Z}(\mathcal{C}, L, \tau') $$

is indeed invariant under the above homotopies. This is sufficient to conclude the proof of Theorem 19.17. □

We give more definitions of $Z$ involving non-necessarily homogeneous propagating forms and pseudo-parallelizations in Chapter 21.
<figure><img src="image_159.png" /></figure>
# Chapter 20

## Simultaneous normalization of propagating forms

This chapter is devoted to the proof of Propositions 18.33 and 18.36. As shown in Section 18.7, this is sufficient to prove Theorem 18.6. In this chapter, we use real coefficients for homology and cohomology unless otherwise mentioned.

### 20.1 Sketch

First note that the homogeneous boundary form of Definition 19.13 defined on $\partial C_2(R)$ (or the form $p_\tau^*(\omega_{S^2})$ as in Definition 3.9 when pseudo-parallelizations are not involved) is antisymmetric on $\partial C_2(R)$ as in Definition 3.14. So it extends as a closed antisymmetric 2-form $\omega = \omega_0$ on $C_2(R)$ as in Lemma 3.16. Also note that if the restriction of $\omega_I$ to

$$A_I^{(i)} \times (C_1(R_I) \setminus A_{I,3}^{(i)}) \subset C_2(R_I)$$

equals

$$\sum_{j \in g_i} p_1^*(\eta_I(a_j^i)) \wedge p_2^*(\eta_I(z_j^i)) + p_2^*(\omega(p^i))$$

as stated in Proposition 18.36, then the restriction of $\omega_I$ to $A_I^{(i)} \times A_I^{(k)}$ equals

$$\sum_{(j, \ell) \in g_i \times g_k} lk(z_j^i, z_\ell^k) p_{A_I^{(i)}}^*(\eta(a_j^i)) \wedge p_{A_I^{(k)}}^*(\eta(a_\ell^k)),$$

for $k \neq i$, as required in Proposition 18.33.

To arrange the propagating forms $\omega_I$ as in Propositions 18.33 and 18.36, we will first show how to make $\omega$ satisfy the conditions of Proposition 18.36,
with respect to the notation before Proposition 18.36. More precisely, we will prove the following proposition in Subsection 20.2.

**Proposition 20.1.** Let $\tilde{\omega}$ be a propagating form of $C_2(R)$ as in Definition 3.11. Its restriction to $\partial C_2(R) \setminus UB_R$ may be expressed as $p_\tau^*(\tilde{\omega}_{S^2})$, for some volume-one form $\tilde{\omega}_{S^2}$ of $S^2$. Let $\tilde{\omega}(p^i)$ (resp. $\tilde{\omega}(p^i)_i$) be a degree-two form on $(C_1(R) \setminus \text{Int}(A^{(i)}))$ that satisfies the same properties as the form $\omega(p^i)$ (introduced before Proposition 18.36) except that it restricts to $\partial C_1(R) = S^2$ as $\tilde{\omega}_{S^2}$ (resp. as $-l_{S^2}^*\tilde{\omega}_{S^2}$) instead of the usual volume form $\omega_{S^2}$.<fnref n="1" /> If $\tilde{\omega}$ is antisymmetric, then assume that $\tilde{\omega}(p^i)_i = \tilde{\omega}(p^i)$.

Then there exists a propagating form $\omega$ of $C_2(R)$ such that

1. the form $\omega$ coincides with $\tilde{\omega}$ on $\partial C_2(R)$,

2. for every $i \in \underline{x}$, the restriction of $\omega$ to

$$A^{(i)} \times (C_1(R) \setminus A_3^{(i)}) \subset C_2(R)$$

is equal to

$$\sum_{j=1}^{g_i} p_1^*(\eta(a_j^i)) \wedge p_2^*(\eta(z_j^i)) + p_2^*(\tilde{\omega}(p^i)),$$

where $p_1$ and $p_2$ respectively denote the first and the second projection of $A^{(i)} \times (C_1(R) \setminus A_3^{(i)})$ to $C_1(R)$, and the restriction of $\omega$ to

$$(C_1(R) \setminus A_3^{(i)}) \times A^{(i)} \subset C_2(R)$$

is equal to

$$\sum_{j \in \underline{g_i}} p_1^*(\eta(z_j^i)) \wedge p_2^*(\eta(a_j^i)) - p_1^*(\tilde{\omega}(p^i)_i),$$

3. for every $i \in \underline{x}$, for any $j \in \{1, 2, \dots, g_i\}$, we have

$$\int_{\Sigma(a_j^i) \times p^i} \omega = 0 \quad \text{and} \quad \int_{p^i \times \Sigma(a_j^i)} \omega = 0,$$

where $p^i \in \partial A^{(i)}$ and $\partial \Sigma(a_j^i) \subset \{4\} \times \partial A^{(i)}$, and

4. $\omega$ is antisymmetric if $\tilde{\omega}$ is.

<footnote n="1">¹In our proof of Propositions 18.33 and 18.36, we use Proposition 20.1 only when $\tilde{\omega}_{S^2} = \omega_{S^2}$ and $\tilde{\omega}(p^i) = \tilde{\omega}(p^i)_i = \omega(p^i)$, but the general statement is useful in other related work.</footnote>
Assume that Proposition 20.1 is proved. This is the goal of Subsection 20.2. Recall that we will use Proposition 20.1 only when $\tilde{\omega}_{S^2} = \omega_{S^2}$ and $\tilde{\omega}(p^i) = \tilde{\omega}(p^i)_i = \omega(p^i)$. Also recall that the $\eta(a_j^i)$ are defined both in $A^{(i)}$ and $A^{(i)'}$ and they are identical near $\partial A^{(i)}$ and $\partial A^{(i)'}$. Finally recall that $\omega(p^i)$ is supported outside $\cup_{k \in \underline{x}} \text{Int}(A^{(k)})$ and the $\eta(z_j^i)$ restrict to the $A^{(k)}$ as a combination of $\eta(a_\ell^k)$ (fixed by the linking numbers). When changing some $A^{(i)}$ into some $A^{(i)'}$ with the same Lagrangian, it is easy to change the restrictions of $\omega$ inside the parts $(A^{(r)} \times (C_1(R) \setminus A_3^{(r)}) \text{ or } (C_1(R) \setminus A_3^{(r)}) \times A^{(r)} \subset C_2(R))$ mentioned in the statement of Proposition 20.1. Indeed, all the forms $\eta(a_j^i)$, $\eta(z_j^i)$, and $\omega(p^i)$ can be defined on the parts of the $R_I$ where they are needed so that these forms coincide with each other whenever it makes sense, and so that they have the properties that were required for $R$. Define $\omega_0(R_I)$ on

$$D(\omega_0(R_I)) = \left( C_2(R_I) \setminus \left( \cup_{i \in I} p_b^{-1} \left( (A_{-1}^{(i)'} \times A_3^{(i)'} \cup (A_3^{(i)'} \times A_{-1}^{(i)'} \right) \right) \right) \cup p_b^{-1}(\text{diag}(\check{R}_I)) \right)$$

so that

1. we have $\omega_0(R_I) = \omega$ on $C_2(R \setminus (\cup_{i \in I} A_{-1}^{(i)'}))$,

2. when $i \in I$, we have

$$\omega_0(R_I) = \sum_{i=1}^{g_I} p_1^* (\eta(a_j^i)) \wedge p_2^* (\eta(z_j^i)) + p_2^* (\omega(p^i))$$
on $A^{(i)'} \times (\check{R}_I \setminus A_3^{(i)'})$,

3. when $i \in I$, we have $\omega_0(R_I) = -\iota^*(\omega_0(R_I))$ on $(\check{R}_I \setminus A_3^{(i)'} \times A^{(i)'})$, and

4. on $\partial C_2(R_I)$, the form $\omega_0(R_I)$ coincides with the homogeneous boundary form $\omega(\tau_I, F)$ of Definition 19.13, for a map $F$ that is the same for all $I \subseteq \underline{x}$.

Note that this definition is consistent.

$$\text{Set } R_i = R_{\{i\}} \text{ and } D_A(\omega_0(R_i)) = C_2(A_4^{(i)'} \cap D(\omega_0(R_i))).$$

**Lemma 20.2.** With the above notation, for any $i \in \underline{x}$, the cohomology class of $\omega_0(R_i)$ vanishes on the kernel of the map

$$H_2(D_A(\omega_0(R_i))) \to H_2(C_2(A_4^{(i)'}))$$

induced by the inclusion.
This lemma was surprisingly difficult to prove for me. It will be proved in Subsection 20.3. Assume it for the moment. Then (the cohomology class of) $\omega_0(R_i)$ is in the image of the natural map

$$H^2(C_2(A_4^{(i)'})) \to H^2(D_A(\omega_0(R_i))).$$

So $\omega_0(R_i)$ extends to a closed form $\omega_1(R_i)$ on $C_2(A_4^{(i)'})$. Change this form to $\omega_{\{i\}} = \frac{1}{2}(\omega_1(R_i) - i^*(\omega_1(R_i)))$ to get an antisymmetric homogeneous propagating form of $(C_2(R_i), \tau_{\{i\}})$. For any $I \subseteq \underline{x}$, define

$$\omega_I = \begin{cases} \omega_0(R_I) & \text{on } C_2(R_I) \setminus (\cup_{i \in I} p_b^{-1}((A_{-1}^{(i)'} \times A_4^{(i)'}) \cup (A_4^{(i)'} \times A_{-1}^{(i)'}))) \\ \omega_{\{i\}} & \text{on } C_2(A_4^{(i)'}) \text{ for } i \in I. \end{cases}$$

This definition is consistent since the $C_2(A_4^{(i)'})$ do not intersect. The forms $\omega_I$ satisfy the properties of Proposition 18.33. In order to finish the proof of Proposition 18.36, up to Proposition 20.1 and Lemma 20.2, let us prove that $\int_{\Sigma'(a_j^i) \times p^i} \omega_{\{i\}} = 0$. Note

$$\int_{\Sigma'(a_j^i) \times p^i} \omega_{\{i\}} = \int_{\Sigma'(a_j^i) \times (p^i \times \{4\})} \omega_{\{i\}} + \int_{\partial \Sigma'(a_j^i) \times (p^i \times [0,4])} \omega_{\{i\}}.$$

The same formula applied to $\Sigma(a_j^i)$ instead of $\Sigma'(a_j^i)$ yields

$$0 = \int_{\Sigma(a_j^i) \times (p^i \times \{4\})} \omega + \int_{\partial \Sigma(a_j^i) \times (p^i \times [0,4])} \omega.$$

The prescribed behavior of the forms on $A_I^{(i)} \times (\tilde{R}_I \setminus A_{I3}^{(i)})$ implies

$$\int_{(\Sigma'(a_j^i) \cap A_0^{(i)'})(p^i \times \{4\})} \omega_{\{i\}} = \int_{(\Sigma(a_j^i) \cap A_0^{(i)}) (p^i \times \{4\})} \omega = 0.$$

Since $\omega$ and $\omega_{\{i\}}$ coincide on $C_2(A_4^{(i)} \setminus A_{-1}^{(i)})$, we also have

$$\int ((-\partial\Sigma(a_j^i) \times [0,4]) \times (p^i \times \{4\})) \cup (\partial\Sigma(a_j^i) \times (p^i \times [0,4])) (\omega_{\{i\}} - \omega) = 0.$$

This shows $\int_{\Sigma'(a_j^i) \times p^i} \omega_{\{i\}} = 0$.

Thus, Proposition 20.1 and Lemma 20.2 imply Propositions 18.33 and 18.36. Their proofs occupy the next two subsections.
## 20.2 Proof of Proposition 20.1

The homology classes of the $(z_j^i \times (4 \times a_k^i))_{(j,k) \in \{1, \dots, g_i\}^2}$ and $(p^i \times \partial C_1(R))$ form a basis of

$$H_2(A^{(i)} \times (C_1(R) \setminus A_3^{(i)})) = (H_1(A^{(i)}) \otimes H_1(R \setminus A^{(i)})) \oplus H_2(C_1(R) \setminus A^{(i)}).$$

According to Lemma 3.12, the evaluation of the cohomology class of any propagating form of $C_2(R)$ at these classes is $lk(z_j^i, (4 \times a_k^i)) = \delta_{kj}$ for the first ones and 1 for the last one. In particular, the form of the statement integrates correctly on this basis.

Let us first prove Proposition 20.1 when $\underline{x} = \{1\}$. Set $A^1 = A$, and forget about the superfluous superscripts 1. Let $\omega_0$ be a propagating two-form of $C_2(R)$ that restricts to $\partial C_2(R) \setminus U B_R$ as $p_\tau^*(\tilde{\omega}_{S^2})$, and let $\omega_b$ be the closed 2-form defined on $(A_1 \times (C_1(R) \setminus \text{Int}(A_2)))$ by the statement (extended naturally). Since this form $\omega_b$ integrates correctly on

$$H_2(A_1 \times (C_1(R) \setminus \text{Int}(A_2))),$$

there exists a one-form $\eta$ on $(A_1 \times (C_1(R) \setminus \text{Int}(A_2)))$ such that $\omega_b = \omega_0 + d\eta$.

This form $\eta$ is closed on $A_1 \times \partial C_1(R)$. Since $H^1(A_1 \times (C_1(R) \setminus \text{Int}(A_2)))$ maps surjectively to $H^1(A_1 \times \partial C_1(R))$, we may extend $\eta$ as a closed one-form $\tilde{\eta}$ on $(A_1 \times (C_1(R) \setminus \text{Int}(A_2)))$. Changing $\eta$ into $(\eta - \tilde{\eta})$ turns $\eta$ to a primitive of $(\omega_b - \omega_0)$ that vanishes on $A_1 \times \partial C_1(R)$. Let $\chi: C_2(R) \to [0, 1]$ be a smooth function supported in $(A_1 \times (C_1(R) \setminus \text{Int}(A_2)))$ and constant with the value 1 on $(A \times (C_1(R) \setminus A_3))$. Set

$$\omega_a = \omega_0 + d\chi\eta.$$

Then $\omega_a$ is a closed form that has the required form on $(A \times (C_1(R) \setminus A_3))$. Furthermore, the restrictions of $\omega_a$ and $\omega_0$ agree on $\partial C_2(R)$ since $d\chi\eta$ vanishes there (because $\eta$ vanishes on $A_1 \times \partial C_1(R)$).

Adding to $\eta$ a combination $\eta_c$ of the closed forms $p_2^*(\eta(z_j))$, which vanish on $A_1 \times \partial C_1(R)$, does not change the above properties, but adds

$$\int_{p \times ([2,4] \times a_j)} d(\chi\eta_c) = \int_{p \times (4 \times a_j)} \eta_c$$

to $\int_{p \times \Sigma(a_j)} \omega_a$. Therefore, since the $p_2^*(\eta(z_j))$ generate the dual of $\mathcal{L}_A$, we may choose $\eta_c$ so that all the $\int_{p \times \Sigma(a_j)} \omega_a$ vanish. After this step, $\omega_a$ is a closed form, which takes the prescribed values on

$$PS_a = \partial C_2(R) \cup (A \times (C_1(R) \setminus A_3)),$$
and such that all the $\int_{p \times \Sigma(a_j)} \omega_a$ vanish. To make $\omega_a$ take the prescribed values on $\iota(PS_a)$ and integrate as required on $\Sigma(a_j) \times p$, we apply similar modifications to $\omega_a$ on the symmetric part $(C_1(R) \setminus \mathrm{Int}(A_2)) \times A_1$. The support of these modifications is disjoint from the support of the previous ones. Therefore, they do not interfere and transform $\omega_a$ into a closed form $\omega_b$ with the following additional properties:

* the form $\omega_b$ has the prescribed form on $(C_1(R) \setminus A_3) \times A$,

* we have $\int_{\Sigma(a_j) \times p} \omega_b = 0$, for all $j = 1, \dots, g_1$.

Thus, the form $\omega = \omega_b$ (resp. $\omega = \frac{\omega_b - \iota^*(\omega_b)}{2}$ if antisymmetry is desired) has all the required properties, and Proposition 20.1 is proved for $\underline{x} = \{1\}$.

We now proceed by induction on $x$. We start with a 2-form $\omega_0$ that satisfies all the hypotheses with $\underline{x} = \underline{1}$ instead of $\underline{x}$. By the first step, we also assume that we have a 2-form $\omega_b$ satisfying all the hypotheses with $\{x\}$ instead of $\underline{x}$, with the enlarged $A_1^{(x)}$ replacing $A^{(x)}$.

Now, we proceed similarly. There exists a one-form $\eta$ on $C_2(R)$ such that $\omega_b = \omega_0 + d\eta$. The exact sequence

$$0 = H^1(C_2(R)) \to H^1(\partial C_2(R)) \to H^2(C_2(R), \partial C_2(R)) \cong H_4(C_2(R)) = 0$$

implies that $H^1(\partial C_2(R))$ is trivial. Therefore, the form $\eta$ is exact on $\partial C_2(R)$. Thus, we may assume that $\eta$ vanishes on $\partial C_2(R)$, which we do. Let

$$\chi: C_2(R) \to [0, 1]$$

be a smooth function supported in $(A_1^{(x)} \times (C_1(R) \setminus \mathrm{Int}(A_2^{(x)})))$ and constant with the value 1 on $(A^{(x)} \times (C_1(R) \setminus A_3^{(x)}))$. Again, we are going to modify $\eta$ by some closed forms so that

$$\omega_a = \omega_0 + d\chi\eta$$

has the prescribed value on

$$PS_a = \partial C_2(R) \cup \left( \bigcup_{k=1}^{x} \left( A^{(k)} \times (C_1(R) \setminus A_3^{(k)}) \right) \right) \cup \left( \bigcup_{k=1}^{x-1} \left( (C_1(R) \setminus A_3^{(k)}) \times A^{(k)} \right) \right).$$

Our form $\omega_a$ is as required everywhere except possibly on

$$(A_1^{(x)} \times (C_1(R) \setminus \mathrm{Int}(A_2^{(x)}))) \setminus (A^{(x)} \times (C_1(R) \setminus A_3^{(x)}))$$
and in particular on the intersection of this domain with the domains where it was normalized previously, which is included in

$$A_1^{(x)} \times (\partial C_1(R) \cup (\cup_{k=1}^{x-1} A^{(k)})).$$

Recall that $\eta$ vanishes on $A_1^{(x)} \times \partial C_1(R)$. Our assumptions also imply that $\eta$ is closed on $A_1^{(x)} \times A^{(k)}$, for any $k < x$. Let us prove that they imply that $\eta$ is exact on $A_1^{(x)} \times A^{(k)}$, for any $k < x$. To do that, it suffices to check the following two assertions.

1. For any $j = 1, \dots, g_x$, we have $\int_{z_j^x \times p^k} \eta = 0$.

2. For any $j = 1, \dots, g_k$, we have $\int_{p^x \times z_j^k} \eta = 0$.

Let us prove the first assertion. Let $\infty(v) \in \partial C_1(R)$, and let $[p^k, \infty(v)]$ be a path from $p^k$ to $\infty(v)$ in $C_1(R)$ that intersects $C$ like the path $[p^k, q^k]$ introduced before Proposition 18.36. Since $\int_{z_j^x \times \infty(v)} \eta = 0$, we have

$$\int_{z_j^x \times p^k} \eta = \int_{\partial(z_j^x \times [p^k, \infty(v)])} \eta = \int_{z_j^x \times [p^k, \infty(v)]} (\omega_b - \omega_0),$$

where $\int_{z_j^x \times [p^k, \infty(v)]} \omega_b = 0$ because the supports of the $\eta(z_j^x)$ do not intersect $[p^k, \infty(v)]$. Let us compute

$$\int_{z_j^x \times [p^k, \infty(v)]} \omega_0 = - \int_{\Sigma(z_j^x) \times \partial[p^k, \infty(v)]} \omega_0 = \int_{\Sigma(z_j^x) \times \{p^k\}} \omega_0.$$

The last integral vanishes because

1. the surface $\Sigma(z_j^x)$ intersects $A_4^{(k)}$ as copies of $\Sigma(a_\ell^k)$,

2. we have $\int_{\Sigma(a_\ell^k) \times p^k} \omega_0 = 0$, thanks to the third condition of Proposition 20.1, and

3. the integral of $\omega_0$ also vanishes on the remaining part of $\Sigma(z_j^x) \times p^k$ since $\omega_0$ is determined on $((C_1(R) \setminus A_4^{(k)}) \times A^{(k)})$ and since the support of $\omega(p^k)$ is disjoint from $\Sigma(z_j^x)$.

Let us prove the second assertion, namely $\int_{p^x \times z_j^k} \eta = 0$ for $j \in g_k$. Again, we have $\int_{\infty(v) \times z_j^k} \eta = 0$ since $\eta$ vanishes on $\partial C_2(R)$. We get

$$\int_{p^x \times z_j^k} \eta = - \int_{[p^x, \infty(v)] \times z_j^k} (\omega_b - \omega_0).$$
The integral $\int_{[p^x, \infty(v)] \times z_j^k} \omega_0$ is zero because of the behavior of $\omega_0$ on $(C_1(R) \setminus A_4^{(k)}) \times A^{(k)}$. We are left with the computation of

$$ \int_{[p^x, \infty(v)] \times z_j^k} \omega_b = \int_{(\partial[p^x, \infty(v)] \times \Sigma(z_j^k))} \omega_b = - \int_{\{p^x\} \times \Sigma(z_j^k)} \omega_b. $$

Again, we know that this integral is zero along the intersection of $\{p^x\} \times \Sigma(z_j^k)$ with $A^{(x)} \times (C_1(R) \setminus A_4^{(x)})$ because $\Sigma(z_j^k)$ does not meet the support of $\omega(p^x)$. We conclude because $\int_{\{p^x\} \times \Sigma(a_\ell^x)} \omega_b = 0$ and because $\Sigma(z_j^k)$ intersects $A_4^{(x)}$ along copies of $\Sigma(a_\ell^x)$.

Since $\eta$ is exact on $A_1^{(x)} \times (\cup_{k=1}^{x-1} A^{(k)})$, we can assume that it vanishes identically there.

Thus, the form $\omega_a$ takes the prescribed values on $A^{(x)} \times (C_1(R) \setminus A_4^{(x)})$, it coincides with $\omega_0$ where $\omega_0$ was prescribed, and it integrates correctly along the $\Sigma(a_\ell^k) \times p^k$ and their symmetric with respect to $\iota$, for $k \neq x$. Let us now modify $\eta$ so that the integrals of $\omega_a$ along the $\{p^x\} \times \Sigma(a_\ell^x)$ vanish for $\ell = 1, \dots, g_x$, too. We do this by adding to $\eta$ a linear combination $(-\eta_c)$ of $p_2^*(\eta(z_j^x))$ that vanishes on the $A_1^{(x)} \times A^{(k)}$, for $k < x$, so that we do not change the properties obtained above. Let $f: H_1(R \setminus \text{Int}(A^{(x)})) \to \mathbb{R}$ be the linear map defined by

$$ f(a_{\ell}^{x}) = \int_{\{p^{x}\} \times \Sigma(a_{\ell}^{x})} \omega_{a}. $$

Set $\eta_c = \sum_{\ell=1}^{g_x} f(a_\ell^x) p_2^*(\eta(z_\ell^x))$. We have $f(y) = \int_{p^x \times y} \eta_c$ for any $y \in \mathcal{L}_{A^{(x)}}$. Fix $k < x$. For any $j \in g_k$, we have

$$ \begin{aligned} f(z_j^k) &= \sum_{\ell=1}^{g_x} lk(z_j^k, z_\ell^x) f(a_\ell^x) = \int_{\{p^x\} \times \Sigma(z_j^k)} \omega_a \\ &= \int_{\{\infty(v)\} \times \Sigma(z_j^k)} \omega_a - \int_{[p^x, \infty(v)] \times z_j^k} \omega_a = 0. \end{aligned} $$

The restriction of $\eta_c$ to $A_1^{(x)} \times A_k$ may be expressed as

$$ \sum_{\ell=1}^{g_x} f(a_{\ell}^{x}) \sum_{j=1}^{g_k} lk(z_{j}^{k}, z_{\ell}^{x}) p_{2}^{*}(\eta(a_{j}^{k})) = \sum_{j=1}^{g_k} f(z_{j}^{k}) p_{2}^{*}(\eta(a_{j}^{k})) = 0. $$

So $\eta_c$ vanishes on the $A_1^{(x)} \times A^{(k)}$ for any $k < x$. Changing $\eta$ into $(\eta - \eta_c)$ does not change $\omega_a$ on the prescribed set but removes $\int_{\{p^x\} \times \Sigma(a_\ell^x)} d\chi\eta_c = \int_{\{p^x\} \times (4 \times a_\ell^x)} \eta_c = f(a_\ell^x)$ from $\int_{\{p^x\} \times \Sigma(a_\ell^x)} \omega_a$, which becomes 0.
After this step, $\omega_a$ is a closed form taking the prescribed values on $PS_a$ such that the integrals of $\omega_a$ along the $(\{p^x\} \times \Sigma(a_\ell^x))$ vanish for $\ell = 1, \dots, g_i$. To make $\omega_a$ take the prescribed values on $\iota(PS_a)$, we apply similar modifications to $\omega_a$ on the symmetric part $(C_1(R) \setminus \tilde{A}_2^{(x)}) \times A_1^{(x)}$. Again, the support of these modifications is disjoint from the support of the previous ones. Thus, they do not interfere. They transform $\omega_a$ to a closed form $\omega_c$ with the following additional properties.

* The form $\omega_c$ has the prescribed form on $(C_1(R) \setminus A_3^{(x)}) \times A^{(x)}$.

* We have $\int_{\Sigma(a_j^x) \times p^x} \omega_c = 0$ for all $j = 1, \dots, g_x$.

Now, the form $\omega = \omega_c$ (resp. $\omega = \frac{\omega_c - \iota^*(\omega_c)}{2}$ if antisymmetry is desired) has all the required properties, and Proposition 20.1 is proved. $\square$

<figure><img src="image_160.png" /></figure>

## 20.3 Proof of Lemma 20.2

In this section, we conclude the proofs of Propositions 18.33 and 18.36 by proving Lemma 20.2. We first state some homological lemmas.

**Lemma 20.3.** Let $S$ be a closed (oriented) surface. Let $S$ and $S^+$ be two copies of $S$. Let $(c_i)_{i \in 2g}$ and $(c_i^*)_{i \in 2g}$ be two dual bases of $H_1(S; \mathbb{Z})$ such that $\langle c_i, c_j^* \rangle_S = \delta_{ij}$. Let $*$ be a point of $\bar{S}$. Set $\mathrm{diag}(S \times S^+) = \{(x, x^+) : x \in S\}$. We have the following equality in $H_2(S \times S^+; \mathbb{Z})$:

$$ [\mathrm{diag}(S \times S^+)] = [* \times S^+] + [S \times *^+] + \sum_{i=1}^{2g} [c_i \times c_i^{*+}]. $$

**PROOF:** We have

$$ H_2(S \times S^+; \mathbb{Z}) = \mathbb{Z}[* \times S^+] \oplus \mathbb{Z}[S \times *^+] \oplus \bigoplus_{(i,j) \in \underline{2g^2}} \mathbb{Z}[c_i \times c_j^{*+}]. $$

The dual basis of the above basis with respect to the intersection form is

$$ ([S \times *^+], [* \times S^+], ([c_i^* \times c_j^+])_{(i,j) \in \underline{2g^2}}). $$

To get the coordinates of $[\mathrm{diag}(S \times S^+)]$ with respect to the first basis, we compute its intersection numbers with the second one. We have

$$ \langle [\mathrm{diag}(S \times S^+)], [c_i^* \times c_i^+] \rangle = \pm 1. $$
The tangent space to $\mathrm{diag}(S \times S^+)$ is naturally parametrized by $(u_i, v_i^*, u_i, v_i^*)$, and the tangent space to $[c_i^* \times c_i^+]$ is naturally parametrized by $(0, w_i^*, x_i, 0)$. So the intersection sign is the sign of the permutation

$$ (u, v, w, x) \mapsto (u, w, x, v), $$

which is +1.

<figure><img src="image_161.png" /></figure>

**Lemma 20.4.** Let $\Sigma$ be a connected compact oriented surface with one boundary component $J(S^1)$ equipped with a basepoint $* = J(1)$. Let $(c_i)_{i \in 2g}$ and $(c_i^*)_{i \in 2g}$ be two dual bases of $H_1(\Sigma; \mathbb{Z})$ such that $\langle c_i, c_j^* \rangle = \delta_{ij}$. Let $\Sigma$ and $\Sigma^+$ be two copies of $\Sigma$. Set $J^+ = J^+(S^1) = \partial\Sigma^+$. Define the subspaces $J \times_{*,\le} J^+$ and $J \times_{*,\ge} J^+$ of $J \times J^+$ to be

$$ J \times_{*,\le} J^+ = \{ (J(\exp(2i\pi t)), J(\exp(2i\pi u))) : (t, u) \in [0, 1]^2, t \le u \} $$

and

$$ J \times_{*,\ge} J^+ = \{ (J(\exp(2i\pi t)), J(\exp(2i\pi u))) : (t, u) \in [0, 1]^2, t \ge u \}. $$

Let $\mathrm{diag}(\Sigma \times \Sigma^+)$ be the subspace $\{(x, x) : x \in \Sigma\}$ of $\Sigma \times \Sigma^+$. Then the chains

$$ C_{*,\le}(\Sigma, \Sigma^+) = \mathrm{diag}(\Sigma \times \Sigma^+) - * \times \Sigma^+ - \Sigma \times *^+ - J \times_{*,\le} J^+ $$

and

$$ C_{*,\ge}(\Sigma, \Sigma^+) = \mathrm{diag}(\Sigma \times \Sigma^+) - * \times \Sigma^+ - \Sigma \times *^+ + J \times_{*,\ge} J^+ $$

are cycles, and we have

$$ [C_{*,\le}(\Sigma, \Sigma^+)] = [C_{*,\ge}(\Sigma, \Sigma^+)] = \sum_{i=1}^{2g} [c_i \times c_i^{*+}] $$

in $H_2(\Sigma \times \Sigma^+; \mathbb{Z})$.

**PROOF:** Since $\partial(J \times_{*,\le} J^+) = \mathrm{diag}(J \times J^+) - * \times J^+ - J \times *^+$, the chain $C_{*,\le}(\Sigma, \Sigma^+)$ is a cycle. Consider the closed surface $S$ obtained from $\Sigma$ by gluing a disk $D$ along $J$. According to Lemma 20.3, we have

$$ [\mathrm{diag}(S \times S^+)] = [* \times S^+] + [S \times *^+] + \sum_{i=1}^{2g} [c_i \times c_i^{*+}] $$
in $H_2(S \times S^+; \mathbb{Z})$. This implies

$$[C_{*,\leq}(\Sigma, \Sigma^+) - C_{*,\leq}(-D, (-D)^+)] = \sum_{i=1}^{2g} [c_i \times c_i^{*+}]$$

in $H_2(S \times S^+; \mathbb{Z})$. Since the cycle $C_{*,\leq}(-D, (-D)^+)$ lies in $D \times D^+$, it is null-homologous there. Since $H_2(\Sigma \times \Sigma^+; \mathbb{Z})$ injects naturally into $H_2(S \times S^+; \mathbb{Z})$, we deduce that $[C_{*,\leq}(\Sigma, \Sigma^+)] = \sum_{i=1}^{2g} [c_i \times c_i^{*+}]$ in $H_2(\Sigma \times \Sigma^+; \mathbb{Z})$. The proof for $C_{*,\geq}(\Sigma, \Sigma^+)$ is similar. □

Consider a rational homology handlebody $A$ with a collar $[-4, 0] \times \partial A$ of its boundary. For $s \in [-4, 0]$, recall $A_s = A \setminus ([s, 0] \times \partial A)$ and $\partial A_s = \{s\} \times \partial A$. Fix pairwise disjoint simple closed curves $(a_i)_{i=1,...,g_A}$ and pairwise disjoint simple closed curves $(z_i)_{i=1,...,g_A}$ on $\partial A$ such that $\mathcal{L}_A = \bigoplus_{i=1}^{g_A}[a_i]$ and $\langle a_i, z_j \rangle_{\partial A} = \delta_{ij}$.

Consider a curve $a$ of $\partial A$ disjoint from the curves $a_i$. The class of $a$ is in $\mathcal{L}_A$. Let $k \in \mathbb{N} \setminus \{0\}$ be its order $H_1(A; \mathbb{Z})$. Let $\Sigma = k\Sigma(a)$ be a surface of $A$ immersed in $A$ bounded by $ka$. Assume that $\Sigma$ intersects $[-1, 0] \times \partial A$ as $k$ copies of $[-1, 0] \times a$, the interior $\text{Int}(A_{-1})$ of $A_{-1}$ as an embedded surface, and $\text{Int}(A_{-1}) \setminus A_{-2}$ as $k$ disjoint annuli. See the thick part of Figure 20.1. For $s \in [-2, 0]$, set $\Sigma_s = \Sigma \cap A_s$.

**Lemma 20.5.** With the above notation, let $(c_i)_{i=1,...,2g}$ and $(c_i^*)_{i=1,...,2g}$ be two dual bases of $H_1(\Sigma_{-2}; \mathbb{Z})/H_1(\partial\Sigma_{-2}; \mathbb{Z})$ such that $\langle c_i, c_j^* \rangle = \delta_{ij}$. Represent $(c_i)_{i=1,...,2g}$ and $(c_i^*)_{i=1,...,2g}$ by curves $(c_i)_{i=1,...,2g}$ and $(c_i^*)_{i=1,...,2g}$ of $\Sigma_{-2}$. Let $\Sigma_{-2} \times [-1, 1]$ denote a tubular neighborhood of $\Sigma_{-2} = \Sigma_{-2} \times \{0\}$ in $A_{-2}$. For a curve $\sigma$ of $\Sigma_{-2}$, the curve $\sigma \times \{1\}$ is denoted by $\sigma^+$.

Then $\sum_{i=1}^{2g} c_i \times c_i^*$ is homologous to $\sum_{(j,\ell) \in g_A^2 \setminus \text{diag}} \langle \Sigma, \Sigma(a_j), \Sigma(a_\ell) \rangle_A z_j \times z_\ell$ in $A^2$. Furthermore, the sum $\sum_{i=1}^{2g} c_i \times c_i^{*+}$ is homologous to

$$\sum_{(j,\ell) \in g_A^2 \setminus \text{diag}} \langle \Sigma, \Sigma(a_j), \Sigma(a_\ell) \rangle_A z_j \times z_\ell - gUA|_*$$

in $C_2(A)$.

**PROOF:** Assume that $\Sigma$ and the $\Sigma(a_j)$ are transverse to each other. For $(j, \ell) \in \{1, ..., g_A\}^2$, set $\gamma_{\Sigma j} = \Sigma \cap \Sigma(a_j)$ and $\gamma_{\Sigma \ell} = \Sigma \cap \Sigma(a_\ell)$. If $j \neq \ell$, also set $\gamma_{j\ell} = \Sigma(a_j) \cap \Sigma(a_\ell)$. We have

$$c_i = \sum_{j=1}^{g_A} \langle c_i, \Sigma(a_j) \rangle_A z_j = \sum_{j=1}^{g_A} \langle c_i, \gamma_{\Sigma j} \rangle_{\Sigma} z_j \quad \text{and} \quad c_i^* = \sum_{\ell=1}^{g_A} \langle c_i^*, \gamma_{\Sigma \ell} \rangle_{\Sigma} z_\ell$$
in $H_1(A)$. This implies

$$c_i \times c_i^* = \sum_{(j,l) \in g_A^2} \langle c_i, \gamma_{\Sigma j} \rangle_{\Sigma} \langle c_i^*, \gamma_{\Sigma l} \rangle_{\Sigma} z_j \times z_l$$

in $H_2(A^2)$. On the other hand, we have

$$\gamma_{\Sigma j} = \sum_{i=1}^{2g} \langle c_i, \gamma_{\Sigma j} \rangle_{\Sigma} c_i^* \quad \text{and} \quad \gamma_{\Sigma l} = - \sum_{i=1}^{2g} \langle c_i^*, \gamma_{\Sigma l} \rangle_{\Sigma} c_i$$

in $H_1(\Sigma_{-2})/H_1(\partial\Sigma_{-2})$. This implies

$$\langle \gamma_{\Sigma j}, \gamma_{\Sigma l} \rangle_{\Sigma} = \sum_{i=1}^{2g} \langle c_i, \gamma_{\Sigma j} \rangle_{\Sigma} \langle c_i^*, \gamma_{\Sigma l} \rangle_{\Sigma}.$$

In particular, we have

$$\sum_{i=1}^{2g} \langle c_i, \gamma_{\Sigma j} \rangle_{\Sigma} \langle c_i^*, \gamma_{\Sigma j} \rangle_{\Sigma} = 0$$

for any $j \in \{1, \dots, g_A\}$. If $j \neq l$, then $\langle \Sigma, \Sigma(a_j), \Sigma(a_l) \rangle_A = \langle \gamma_{\Sigma j}, \gamma_{\Sigma l} \rangle_{\Sigma}$. So $\alpha = \sum_{i=1}^{2g} c_i \times c_i^{*+}$ is homologous to $\sum_{(j,l) \in g_A^2 \setminus \text{diag}} \langle \Sigma, \Sigma(a_j), \Sigma(a_l) \rangle_A z_j \times z_l$ as announced.

Let us now compute the homology class $[\alpha]$ of $\alpha$ in $H_2(C_2(A))$. For a curve $\sigma$ of $\Sigma_{-2}$, both $a \times \sigma^+$ and $\sigma \times a^+$ are null-homologous in $C_2(A)$. Hence, the class $[\alpha]$ depends only on the class of $\sum_{i=1}^{2g} c_i \otimes c_i^{*+}$ in $H_1(\Sigma_{-2})/H_1(\partial\Sigma_{-2}) \otimes H_1(\Sigma_{-2}^+) / H_1(\partial\Sigma_{-2}^+)$. This class is determined by the property that for any two closed curves $e$ and $f$ of $\Sigma$, we have $\langle e \times f^+, \alpha \rangle_{\Sigma \times \Sigma^+} = -\langle e, f \rangle_{\Sigma}$. So the class $[\alpha] \in H_2(C_2(A))$ is independent of the dual bases $(c_i)$ and $(c_i^*)$. In particular, we have $[\alpha] = [\sum_{i=1}^{2g} c_i^* \times (-c_i^*)]$. Set

$$\beta = \sum_{(j,l) \in g_A^2 \setminus \text{diag}} \langle \Sigma, \Sigma(a_j), \Sigma(a_l) \rangle_A z_j \times z_l - gUA|_*.$$

The previous computation tells us that the difference $[\alpha - \beta]$ is a rational multiple of $[UA|_*]$ in $H_2(C_2(A))$. To evaluate this multiple, we embed $A$ in a rational homology ball obtained from $A$ by adding thickened disks along neighborhoods of the $z_i$. Embed this rational homology ball in a rational homology sphere $R$. We get

$$[\alpha - \beta] = \frac{1}{2} \sum_{i=1}^{2g} (lk_R(c_i, c_i^{*+}) - lk_R(c_i^{*-}, c_i))[UA|_*] \\ - \sum_{(j,l) \in \{1, \dots, g_A\}^2 \setminus \text{diag}} \langle \Sigma, \Sigma(a_j), \Sigma(a_l) \rangle_A lk_R(z_j, z_l)[UA|_*]$$
in $H_2(C_2(R); \mathbb{R})$. Since $lk_R(z_j, z_\ell) = 0$, the second row vanishes and we obtain $[\alpha - \beta] = -g[UA|_*]$ in $H_2(C_2(R); \mathbb{R})$. Since $[UA|_*] \neq 0$ in $H_2(C_2(R); \mathbb{R})$, this equality also holds in $H_2(C_2(A); \mathbb{R})$. $\square$

We now define a cycle $F^2(\Sigma(a))$ of $\partial C_2(A)$ associated to the surface $\Sigma = k\Sigma(a)$ introduced before Lemma 20.5. Let $(a \times [-1, 1])$ be a tubular neighborhood of $a$ in $\partial A$. Let $p(a) \in a$. View $a$ as the image of a map $a: [0, 1] \to a$ such that $a(0) = a(1) = p(a)$. Set

$$ \Sigma^+ = \Sigma_{-1} \cup k\{(t-1, a(\alpha), t) : (t, \alpha) \in [0, 1]^2\}. $$

So $\partial\Sigma^+ = ka^+$, where $a^+ = a \times \{1\}$. Set

$$ p(a)^+ = (p(a), 1) = (0, p(a), 1) \in a \times [-1, 1] \subset (\partial A = \{0\} \times \partial A). $$

Recall $a \times_{p(a), \ge} a^+ = \{((a(v), 0), (a(w), +1)) : (v, w) \in [0, 1]^2, v \ge w\}$. Let $T(a)$ be the closure of $\{((a(v), 0), (a(v), t)) : (t, v) \in ]0, 1] \times [0, 1]\}$ (oriented by $(t, v)$) in $\partial C_2(A)$. Let $s_+(\Sigma)$ be the positive normal section of $U(A)|_\Sigma$. Let $g$ be the genus of $\Sigma$. Set $e(\Sigma(a) = \frac{1}{k}\Sigma) = \frac{1}{k}(g + k - 1)$. We have

$$ e(\Sigma(a)) = \frac{-\chi(\Sigma)}{2k} + \frac{1}{2}. $$

**Lemma 20.6.** With the above notation, the chain

$$ F^2(\Sigma(a)) = \frac{1}{k}s_+(\Sigma) + T(a) - \left(p(a) \times \frac{1}{k}\Sigma^+\right) - \left(\frac{1}{k}\Sigma \times p(a)^+\right) + \left(a \times_{p(a), \ge} a^+\right) \\ + e(\Sigma(a))[UA|_*] - \sum_{(j, \ell) \in \{1, \dots, gA\}^2 \setminus \text{diag}} \langle \Sigma(a), \Sigma(a_j), \Sigma(a_\ell) \rangle z_j \times z_\ell $$

is a cycle, and it is null-homologous in $C_2(A)$.

**PROOF:** For $k=1$ (when we are dealing with integral homology handlebodies, for example), it is a direct consequence of Lemma 20.4 and Lemma 20.5 above. Let us now focus on the case $k > 1$. Observe that $F^2(\Sigma(a))$ is a cycle. Without loss of generality, assume

$$ \Sigma \cap ([-2, -1] \times \partial A) = \left\{ (t-2, a(\alpha), (1-t)\frac{j-1}{k}) : (t, \alpha) \in [0, 1]^2, j \in \underline{k} \right\}. $$

For the proof, we change $\Sigma^+$ to a surface (still denoted by $\Sigma^+$) with the same boundary as follows.
Recall $\Sigma_{-2} = \Sigma \cap A_{-2}$. We have $\partial\Sigma_{-2} = \cup_{j=1}^k (\{-2\} \times a \times \{\frac{j-1}{k}\})$. Let $\Sigma_{-2}^+$ be a parallel copy of $\Sigma$ on its positive side with boundary $\partial\Sigma_{-2}^+ = \cup_{j=1}^k (\{-2\} \times a \times \{\frac{j-1}{k}\})$. Redefine $\Sigma^+$ so that we have $\Sigma^+ \cap A_{-2} = \Sigma_{-2}^+$,

$$ \Sigma^+ \cap ([-1, 0] \times \partial A) = k([-1, 0] \times a \times \{1\}), $$

and

$$ \Sigma^+ \cap ([-2, -1] \times \partial A) = \{ (t-2, a(\alpha), (1-t)\frac{j-1/2}{k}+t) : (t, \alpha) \in [0, 1]^2, j \in \underline{k} \} $$

as in Figure 20.1, which represents $\Sigma \cap ([-2, 0] \times p(a) \times [-1, 1])$ as the thick lines and $\Sigma^+ \cap ([-2, 0] \times p(a) \times [-1, 1])$ as the thin lines when $k=3$. Observe that this modification of $\Sigma^+$ changes neither the boundary of $\Sigma^+$ nor the class of the resulting cycle $F^2(\Sigma(a))$ in $H_2(C_2(A))$.

<figure><img src="image_162.png" /><figcaption>Figure 20.1: How $\Sigma$ and $\Sigma^+$ intersect ($[-2, 0] \times p(a) \times [-1, 1]$) in the proof of Lemma 20.6.</figcaption></figure>

Let $S$ be the closed surface obtained from $\Sigma_{-2}$ by gluing abstract disks $D_j$ with respective boundaries $\{-2\} \times (-a) \times \{\frac{j-1}{k}\}$ on $\partial\Sigma_{-2}$. Let $S^+$ be similarly obtained from $\Sigma_{-2}^+$ by gluing abstract disks $D_j^+$ with boundaries $\{-2\} \times (-a) \times \{\frac{j-1}{k}\}$ on $\partial\Sigma_{-2}^+$. For $j=1, \dots, k$, set $p_j = (-2, p(a), \frac{j-1}{k}) \in \partial A_{-2}$ and $p_j^+ = (-2, p(a), \frac{j-1}{k})$. Recall the dual bases $(c_i)$ and $(c_i^*)$ of Lemma 20.5. Lemma 20.3 implies that

$$ C(S) = \mathrm{diag}(S \times S^+) - p_1 \times S^+ - S \times p_k^+ - \sum_{i=1}^{2g} c_i \times c_i^{*+} $$

is null-homologous in $H_2(S \times S^+)$. Choose closed representatives $c_i$ of the classes $c_i$ in the interior of $\Sigma_{-2}$ so that $(\Sigma_{-2} \setminus \cup_{i=1}^{2g} c_i)$ is connected. Let $[p_1, p_j]$ denote a path from $p_1$ to $p_j$ in $(\Sigma_{-2} \setminus \cup_{i=1}^{2g} c_i)$. Let $[p_j^+, p_k^+]$ be a path from $p_j^+$ to $p_k^+$ in $(\Sigma_{-2}^+ \setminus \cup_{i=1}^{2g} c_i^+)$. Add to $C(S)$ the null-homologous cycles

$$ \partial(-[p_1, p_j] \times D_j^+) = p_1 \times D_j^+ - p_j \times D_j^+ + [p_1, p_j] \times \partial D_j^+, $$

$$ \partial(D_j \times [p_j^+, p_k^+]) = D_j \times p_k^+ - D_j \times p_j^+ + \partial D_j \times [p_j^+, p_k^+], $$
and the null-homologous cycles $(-C_{*,\le}(D_j, D_j^+))$ of Lemma 20.4, for $j = 1, \dots, k$. This addition transforms $C(S)$ to the null-homologous cycle

$$C(\Sigma_{-2}) = \operatorname{diag}(\Sigma_{-2} \times \Sigma_{-2}^+) - p_1 \times \Sigma_{-2}^+ - \Sigma_{-2} \times p_k^+ - \sum_{i=1}^{2g} c_i \times c_i^{*+} \\ + \sum_{j=1}^k (\partial D_j \times [p_j^+, p_k^+] + [p_1, p_j] \times \partial D_j^+ + \partial D_j \times_{p(a),\le} \partial D_j^+).$$

Let us deform the cycle $C(\Sigma_{-2})$ continuously in $\Sigma \times \Sigma^+$ to move the level $\{-2\} \times \partial A$ to the level $\{0\} \times \partial A$. During such a deformation, when $s$ tends to $(-1)$, the path $[\{s\} \times p_j^+, \{s\} \times p_k^+]$ becomes a loop $[p_j^+, p_k^+]_{-1}$ on $\Sigma_{-1}^+ = \Sigma^+ \cap A_{-1}$, and the path $[\{s\} \times p_1, \{s\} \times p_j]$ becomes a loop $[p_1, p_j]_{-1}$ on $\Sigma_{-1}$. Thus the cycle $C(\Sigma_{-2})$ can be naturally deformed in $\Sigma \times \Sigma^+$ to the following still null-homologous cycle:

$$C(\Sigma) = \operatorname{diag}(\Sigma \times \Sigma^+) - p(a) \times \Sigma^+ - \Sigma \times p(a)^+ - \sum_{i=1}^{2g} c_i \times c_i^{*+} \\ + \sum_{j=1}^{k} ((-a) \times [p_j^+, p_k^+]_{-1} + [p_1, p_j]_{-1} \times (-a^+)) + k (a \times_{p(a),\ge} a^+).$$

Since $a$ bounds $\frac{1}{k}\Sigma$, the cycle $(-a) \times [p_j^+, p_k^+]_{-1}$ is homologous to the cycle $\langle -\frac{1}{k}\Sigma, [p_j^+, p_k^+]_{-1} \rangle_A U A|_*$, in $C_2(A)$. Similarly, $[p_1, p_j]_{-1} \times (-a^+)$ is homologous to $\langle -\frac{1}{k}\Sigma^+, [p_1, p_j]_{-1} \rangle_A U A|_*$, in $C_2(A)$. Intersections occur where $\Sigma$ and $\Sigma^+$ intersect, in $([-2, -1] \times \partial A)$, as shown in Figure 20.2, where the positive normal to $\Sigma$ goes from left to right.

<figure><img src="image_163.png" /><figcaption>Figure 20.2: The intersection $\Sigma \cap \Sigma^+$ and the loop $[p_j^+, p_k^+]_{-1}$ ($j = 1, k = 3$)</figcaption></figure>

For any $j \in \underline{k}$, we have

$$\langle -\frac{1}{k}\Sigma, [p_j^+, p_k^+]_{-1} \rangle_A = \frac{k-j}{k} \quad \text{and} \quad \langle -\frac{1}{k}\Sigma^+, [p_1, p_j]_{-1} \rangle_A = \frac{j-1}{k}.$$

Therefore, Lemma 20.5 implies that the null-homologous cycle $C(\Sigma)$ is homologous to

$$\operatorname{diag}(\Sigma \times \Sigma^+) - p(a) \times \Sigma^+ - \Sigma \times p(a)^+ - \sum_{(j,\ell) \in \{1, \dots, g_A\}^2 \setminus \operatorname{diag}} \langle \Sigma, \Sigma(a_j), \Sigma(a_\ell) \rangle z_j \times z_\ell \\ + g U A|_* + (k-1) U A|_* + k (a \times_{p(a),\ge} a^+).$$
This cycle is naturally homologous to $kF^2(\Sigma(a))$, which is therefore homologous to zero. □

**Lemma 20.7.** If A is a rational homology handlebody such that $H_1(A) = \bigoplus_{j=1}^{g(A)} \mathbb{R}[z_j]$, then we have

$$H_3(C_2(A)) = \bigoplus_{j=1}^{g(A)} \mathbb{R}[UA|_{z_j}].$$

**PROOF:** The configuration spaces $C_2(A)$ and $C_2(\mathring{A})$ have the same homotopy type, which is the homotopy type of $\mathring{A}^2 \setminus \text{diag}$. We have $H_3(\mathring{A}^2) = H_4(\mathring{A}^2) = 0$ and $H_4(\mathring{A}^2, \mathring{A}^2 \setminus \text{diag}) \cong H_4(\mathring{A}^2 \times B^3, \mathring{A}^2 \times S^2) = \bigoplus_{j=1}^{g(A)} \mathbb{R}[z_j \times B^3]$. □

**Lemma 20.8.** Let $i \in \underline{x}$. For any $j \in \underline{g_i}$, assume that the chains $\Sigma(a_j^i)$ and $\Sigma'(a_j^i)$ defined before Proposition 18.36 intersect $[-1, 4] \times \partial A^{(i)}$ as $[-1, 4] \times a_j^i$. Also assume that they may be respectively expressed as $\frac{1}{k}\Sigma'$ and $\frac{1}{k}\Sigma$, for immersed surfaces $\Sigma$ and $\Sigma'$, which respectively intersect $\text{Int}(A_{-1}^{(i)})$ and $\text{Int}(A_{-1}^{(i)})$ as embedded surfaces (as before Lemma 20.5). Fix $p(a_j^i)$ on $\{4\} \times a_j^i$. Then the classes of the cycles $F^2(\Sigma'(a_j^i))$ of $\partial C_2(A_4^{(i)})$ defined in Lemma 20.6, for $j \in \underline{g_i}$ generate the kernel of the map induced by the inclusion

$$H_2(D_A(\omega_0(R_i))) \to H_2(C_2(A_4^{(i)})).$$

for the domain $D_A(\omega_0(R_i))$ defined before Lemma 20.2.

**PROOF:** First note that $D_A(\omega_0(R_i))$ is homotopically equivalent by retraction to $\partial C_2(A_4^{(i)})$. The cycles $F^2(\Sigma'(a_j^i))$ lie in $\partial C_2(A_4^{(i)})$ and bound chains $G^3(a_j^i)$ in $C_2(A_4^{(i)})$ according to Lemma 20.6. These chains can be assumed to be transverse to the boundary. So they satisfy

$$\langle [G^3(a_j^i)], [UA^{(i)'}|_{z_k^i}] \rangle_{C_2(A_4^{(i)})} = \pm \langle F^2(\Sigma'(a_j^i)), UA^{(i)'}|_{\{0\} \times z_k^i} \rangle_{\partial C_2(A_4^{(i)})} = \pm \delta_{jk}.$$

Therefore, Poincaré duality and Lemma 20.7 imply

$$H_3(C_2(A_4^{(i)}), \partial C_2(A_4^{(i)})) = \bigoplus_{j=1}^{g_i} \mathbb{R}[G^3(a_j^i)].$$

The boundary map of the long exact sequence associated to the pair $(C_2(A_4^{(i)}), \partial C_2(A_4^{(i)}))$ sends the cycle $[G^3(a_j^i)]$ of $(C_2(A_4^{(i)}), \partial C_2(A_4^{(i)}))$ to $[F^2(\Sigma'(a_j^i))]$. □

**PROOF OF LEMMA 20.2:** According to Lemma 20.8, it suffices to prove

$$\int_{F^2(\Sigma'(a_j^i))} \omega_0(R_i) = 0$$
for any $i \in \underline{x}$, and for any $j \in \underline{g_i}$. Fix $i \in \underline{x}$ and $j \in \underline{g_i}$. Set $a = \{4\} \times a_j^i$. Let $F'$ denote the cycle $F^2(\Sigma'(a))$ of $\partial C_2(A_4^{(i)'})$ associated to $\Sigma' = k'\Sigma'(a)$ and to $p(a) = p(a_j^i)$, and let $F$ denote the cycle $F^2(\Sigma(a))$ of $\partial C_2(A_4^{(i)})$ similarly associated to $\Sigma(a)$ and to $p(a)$. We have

$$F' = T(a) + a \times_{p(a),\ge} a^+ - p(a) \times \frac{1}{k'} \Sigma'^+ - \frac{1}{k'} \Sigma' \times p(a)^+ + \frac{1}{k'} s_+(\Sigma') \\ + e(\Sigma'(a))[UA|_*] - \sum_{(p,q) \in \{1, \dots, g_i\}^2 \setminus \text{diag}} \langle \Sigma'(a), \Sigma'(a_p^i), \Sigma'(a_q^i) \rangle (\{4\} \times z_p^i) \times (\{4\} \times z_q^i)$$

Set $\Sigma'^+ = \Sigma'^+ \cap A'^{(i)}$. The integral of $\omega_0(R_i)$ along

$$-p(a) \times \frac{1}{k'} \Sigma'^+ - \frac{1}{k'} \Sigma' \times p(a)^+$$

is zero because of the prescribed form of $\omega_0(R_i)$ on

$$((\check{R} \setminus A_3^{(i)'} \times A^{(i)'} ) \cup (A^{(i)'} \times (\check{R} \setminus A_3^{(i)'}))).$$

For the form $\omega$ of Proposition 20.1, the same argument implies

$$\int_{-p(a) \times \Sigma'^+ \setminus (a_j^i) - \Sigma^- \setminus (a_j^i) \times p(a)^+} \omega = 0.$$

The part

$$C = T(a) + a \times_{p(a),\ge} a^+ - \left(p(a) \times \frac{1}{k'} (\Sigma'^+ \setminus \Sigma'^- )\right) - \left(\frac{1}{k'} (\Sigma' \setminus \Sigma'^- ) \times p(a)^+\right) \\ + \frac{1}{k'} s_+(\Sigma' \setminus \Sigma'^0)$$

of $F'$ (or $F$) lies in the intersection of $C_2(A_4^{(i)})$ and $C_2(A_4^{(i)'})$, inside which $\omega = \omega_0(R_i)$. So we have

$$\int_C \omega = \int_C \omega_0(R_i).$$

Recall

$$\int_{(\{4\} \times z_p^i) \times (\{4\} \times z_q^i)} \omega = lk(z_p^i, z_q^i)$$

for any $(p, q) \in \{1, \dots, g_i\}^2 \setminus \text{diag}$. Since $lk(z_p^i, z_q^i) = lk(z_q^i, z_p^i)$, and since

$$\langle \Sigma(a), \Sigma(a_p^i), \Sigma(a_q^i) \rangle = - \langle \Sigma(a), \Sigma(a_q^i), \Sigma(a_p^i) \rangle,$$
the integral of $\omega$ along

$$ \sum_{(p,q) \in \{1, \dots, g_i\}^2 \setminus \text{diag}} \langle \Sigma(a), \Sigma(a_p^i), \Sigma(a_q^i) \rangle (\{4\} \times z_p^i) \times (\{4\} \times z_q^i) $$

is equal to zero. The integral of $\omega_0(R_i)$ along

$$ \sum_{(p,q) \in \{1, \dots, g_i\}^2 \setminus \text{diag}} \langle \Sigma'(a), \Sigma'(a_p^i), \Sigma'(a_q^i) \rangle (\{4\} \times z_p^i) \times (\{4\} \times z_q^i) $$

vanishes similarly.

Since $\int_F \omega = 0$, we have

$$ \int_{F'} \omega_0(R_i) = \int_{F'} \omega_0(R_i) - \int_F \omega \\ = \int_{\frac{1}{k'} s_+(\Sigma'_0)} \omega_0(R_i) - \int_{s_+(\Sigma_0(a_j^i))} \omega + e(\Sigma'(a)) - e(\Sigma(a)). $$

Recall that $\tau$ coincides with $\tau_i$ on $[0,4] \times \partial A^{(i)}$. When $\tau$ maps $e_3$ to the positive normal to $\Sigma'_0$ along $\partial\Sigma'_0$, Proposition 19.22 and Lemma 19.19 imply

$$ \int_{\frac{1}{k'} s_+(\Sigma'_0)} \omega_0(R_i) = \frac{1}{2} d(\tau(\cdot \times e_2), \{0\} \times a) + \frac{1}{2k'} \chi(\Sigma'_{-2}). $$

We get

$$ \int_{\frac{1}{k'} s_+(\Sigma'_0)} \omega_0(R_i) + e(\Sigma'(a)) = \frac{1}{2} d(\tau(\cdot \times e_2), \{0\} \times a) + \frac{1}{2} \\ = \int_{s_+(\Sigma_0(a_j^i))} \omega + e(\Sigma(a)) $$

and $\int_{F'} \omega_0(R_i) = 0$. When $\tau$ does not map $e_3$ to the positive normal to $\Sigma'_0$ along $\partial\Sigma'_0$, perform a simultaneous homotopy on $\tau$ and $\tau_i$ to make this happen without changing

$$ \int_{\frac{1}{k'} s_+(\Sigma'_0)} \omega_0(R_i) - \int_{s_+(\Sigma_0(a_j^i))} \omega. $$

Thus, the above proof still implies $\int_{F'} \omega_0(R_i) = 0$. $\square$

<figure><img src="image_164.png" /></figure>
# Chapter 21

## Much more flexible definitions of $Z$

### 21.1 More propagating forms associated to pseudo-parallelizations

In this section, we define nonhomogeneous propagating forms associated with pseudo-parallelizations, and we give more flexible definitions of $Z$ involving these forms. In Section 21.2, we define propagating chains associated with pseudo-parallelizations, which allow discrete computations of $Z$ associated with pseudo-parallelizations. As in Chapter 11 and Section 17.1, the corresponding discrete definition of $Z$ are justified by using nonhomogeneous propagating forms $\epsilon$-dual (as in Definition 11.6) to these propagating chains.

Let $A$ be an oriented 3-manifold with possible boundary, equipped with a pseudo-parallelization $\tilde{\tau} = (N(\gamma); \tau_e, \tau_b)$ as in Definition 19.9. Let $\omega_s$ be a 2-form of $S^2$ invariant under the rotations around the vertical axis such that $\int_{S^2} \omega_s = 1$. Let $\omega_i$ be a 2-form of $S^2$ such that $\int_{S^2} \omega_i = 1$. Let $\eta_{i,s,1}$ be a 1-form of $S^2$ such that $\omega_i = \omega_s + d\eta_{i,s,1}$. Let $\epsilon \in ]0, 1/2[$ be the small positive number of Definitions 19.9 and 19.13. Let $\epsilon_i \in ]0, \epsilon/2[$, and let $k$ be a large integer greater than 3. Let $p(\tau_b)$ denote the projection from $UN(\gamma)$ to $S^2$ induced by $\tau_b$. We have

$$p(\tau_b)(\tau_b(t, c, u; X \in S^2)) = X.$$

**Lemma 21.1.** Under the hypotheses above, there exists a one-form $\eta_{i,s}$ on $U(A)$ such that

$$\bullet \quad \eta_{i,s} = p(\tau_e)^* (\eta_{i,s,1}) \text{ on } U(A \setminus ([a, b - \epsilon_i + \epsilon_i^k] \times \gamma \times [-1, 1])),$$
• $\eta_{i,s} = p(\tau_b)^* (\eta_{i,s,1})$ on $[a, b] \times \gamma \times N(\partial[-1, 1]) \times S^2$,

• $\eta_{i,s} = \frac{1}{2}(p(\tau_b \circ \mathcal{T}_\gamma^{-1})^* (\eta_{i,s,1}) + p(\tau_b \circ F(\gamma, \tau_b)^{-1})^* (\eta_{i,s,1}))$ on $[a, b - \varepsilon_i - \varepsilon_i^k] \times \gamma \times [-1, 1] \times S^2$, and

• $\eta_{i,s}$ pulls back through $p_{[a,b]} \times p_{[-1,1]} \times p(\tau_b)$ on $U(N(\gamma) = [a, b] \times \gamma \times [-1, 1])$.

The following addition is useless in this book but useful for studying equivariant invariants as in [Les13]. Let $c$ and $d$ be two elements of $]-1, 1[$ such that $c < d$. Let $v_3$ denote the projection on the third coordinate in $\mathbb{R}^3$, and let $S_{[c,d]}^2 = \{X \in S^2 : v_3(X) \in ]c, d[\}$. If $\omega_s$ and $\omega_i$ are compactly supported in $S_{[c,d]}^2$, then $\eta_{i,s,1}$ can be chosen so that it is also compactly supported in $S_{[c,d]}^2$, and $\eta_{i,s}$ can be chosen so that it is compactly supported in $[a, b] \times \gamma \times [-1, 1] \times S_{[c,d]}^2$.

**PROOF:** The existence of $\eta_{i,s,1}$ compactly supported in $S_{[c,d]}^2$, when $\omega_s$ and $\omega_i$ are compactly supported in $S_{[c,d]}^2$, comes from the fact that $H_c^2(S_{[c,d]}^2) = \mathbb{R}$, according to Theorem B.3.

Extend $\eta_{i,s}$ to

$$C(\times\gamma) \times S^2 = [b - \varepsilon_i - \varepsilon_i^k, b - \varepsilon_i + \varepsilon_i^k] (\times\gamma) \times [-1 + \frac{\varepsilon}{2}, 1 - \frac{\varepsilon}{2}] \times S^2$$

as follows. Cover $S^2$ by three open spaces $S_{[c,d]}^2$, $N(N) = \{X \in S^2 : v_3(X) > \frac{c+d}{2}\}$, and $N(S) = \{X \in S^2 : v_3(X) < \frac{c+d}{2}\}$. Pick a corresponding partition of unity $(\chi_{S_{[c,d]}^2}, \chi_{N(S)}, \chi_{N(N)})$ of functions respectively compactly supported on $S_{[c,d]}^2$, $N(N)$, and $N(S)$ whose sum is one. Let $S$ stand for $S_{[c,d]}^2$, $N(N)$, or $N(S)$. On the product $C(\times\gamma) \times S$ the form $\eta_{i,s}$ is a combination of basic standard one-forms. Smoothly extend the $\mathbb{R}$-valued coordinate functions of the forms from $(\partial(C(\times\gamma))) \times S$ to $C(\times\gamma) \times S$ in order to obtain $\eta_{i,s}$ with the required properties, so that $\eta_{i,s}$ is compactly supported in $[a, b] \times \gamma \times [-1, 1] \times S_{[c,d]}^2$ when $\eta_{i,s,1}$ is compactly supported in $S_{[c,d]}^2$. $\square$

Recall the notation of Definition 19.13 and extend $\tau_e$ to

$$U([b - \varepsilon_i + \varepsilon_i^k, b] \times \gamma \times [-1, 1]),$$

so that it coincides with $\tau_b$, there.

**Definition 21.2.** Under the hypotheses of the beginning of this section, define $\omega = \omega(\tilde{\tau}, \omega_i, k, \varepsilon_i, \eta_{i,s})$ on $UA$ to satisfy

$$\omega = p(\tau_e)^* (\omega_i) \text{ on } U(A \setminus ([a, b - \varepsilon_i + \varepsilon_i^k] \times \gamma \times [-1, 1])),$$
$$ \omega = \frac{1}{2} \left( p(\tau_b \circ \mathcal{T}_{\gamma}^{-1})^* (\omega_i) + p(\tau_b \circ F(\gamma, \tau_b)^{-1})^* (\omega_i) \right) \\ \text{on } U([a, b - \epsilon_i - \epsilon_i^k] \times \gamma \times [-1, 1]), $$

and

$$ \omega = p(\tau_b)^* (\omega_s) + d\eta_{i,s} \text{ on } U([b - \epsilon_i - 2\epsilon_i^k, b - \epsilon_i + 2\epsilon_i^k] \times \gamma \times [-1, 1]). $$

**Lemma 21.3.** *Definition 21.2 of $\omega(\tilde{\tau}, \omega_i, k, \epsilon_i, \eta_{i,s})$ is consistent. Furthermore, we have $\omega(\tilde{\tau}, \omega_i, k, \epsilon_i, \eta_{i,s}) = \omega(\tilde{\tau}, \omega_s, k, \epsilon_i, 0) + d\eta_{i,s}$ on UA.*

**PROOF:** Any 2-form $\omega_s$ invariant under the rotations around the vertical axis may be expressed as $(\kappa \circ v_3)\omega_{S^2}$. Therefore, when $\eta_{i,s}$ is zero, Lemma 19.14 ensures that the definitions of $\omega(\tilde{\tau}, \omega_s, k, \epsilon_i, 0)$ match on $[b - \epsilon_i - 2\epsilon_i^k, b - \epsilon_i - \epsilon_i^k] \times \gamma \times [-1, 1]$. Conclude by observing that $\omega(\tilde{\tau}, \omega_i, k, \epsilon_i, \eta_{i,s}) = \omega(\tilde{\tau}, \omega_s, k, \epsilon_i, 0) + d\eta_{i,s}$ on $UN(\gamma)$. $\square$

**Definition 21.4** (General boundary form associated with a pseudo-parallelization $\tilde{\tau}$). Let $A$ be an oriented 3-manifold equipped with a pseudo-parallelization $\tilde{\tau}$. A boundary form of $(A, \tilde{\tau})$ is a 2-form on $UA$ that may be expressed as $\omega(\tilde{\tau}, \omega_i, k, \epsilon_i, \eta_{i,s})$ for some $(\omega_i, k, \epsilon_i, \eta_{i,s})$ as in Definition 21.2. When $\eta_{i,s,1}$, $\omega_s$, and $\omega_i$ are compactly supported in $S_{[c,d]}^2$, and when the intersection of the support of $\eta_{i,s}$ with $[b - \epsilon_i - 2\epsilon_i^k, b - \epsilon_i + 2\epsilon_i^k] \times \gamma \times [-1, 1] \times S^2$ is a compact subspace of $[b - \epsilon_i - 2\epsilon_i^k, b - \epsilon_i + 2\epsilon_i^k] \times \gamma \times [-1, 1] \times S_{[c,d]}^2$, the boundary form is said to be adapted to $S_{[c,d]}^2$.

Note that Definition 21.4 coincides with Definition 19.13 of a homogeneous boundary form when $\omega_i = \omega_s = \omega_{S^2}$ and $\eta_{i,s} = 0$.

**Lemma 21.5.** *Under the assumptions of Proposition 19.22, for any boundary form $\omega$ of $(M, \tilde{\tau})$ as in Definition 21.4, we have*

$$ \int_{s_+(\Sigma)} \omega = \frac{1}{2} \chi(\tilde{\tau}(\cdot \times e_2)|_{\partial\Sigma}; \Sigma) $$

and

$$ \int_{s_-(\Sigma)} \omega = -\frac{1}{2} \chi(\tilde{\tau}(\cdot \times e_2)|_{\partial\Sigma}; \Sigma). $$

**PROOF:** Thanks to Lemma 19.20, Lemma 21.5 holds when $\Sigma$ does not meet $N(\gamma)$. Therefore, as in the proof of Proposition 19.22, it suffices to prove Lemma 21.5 when $\Sigma$ is a meridian of the link $\gamma$ of the pseudo-parallelization $\tilde{\tau}$. Let us treat this case. When $\omega_i = \omega_s$ and $\eta_{i,s} = 0$, the proof of Proposition 19.22 applies. In general, let $\tilde{\omega}$ be the form obtained from $\omega$ by changing $\omega_i$ to $\omega_s$ and $\eta_{i,s}$ to 0. The form $\omega$ may be written as $\tilde{\omega} + d\eta_{i,s}$ on
UN(γ), where η_{i,s} is expressed as p(τ_e)^* (η_{i,s,1}) along s_+(Σ)|_{∂Σ}. We have
∫_{s_+(Σ)} ω - ∫_{s_+(Σ)} ˜ω = ∫_{s_+(Σ)|_{∂Σ}} p(τ_e)^* (η_{i,s,1}). This is zero since p(τ_e) maps
s_+(Σ)|_{∂Σ} to a point. We similarly get ∫_{s_-(Σ)} ω = ∫_{s_-(Σ)} ˜ω. □

Theorem 19.25 generalizes as follows to these boundary forms.

**Theorem 21.6.** Let A be a compact 3-manifold that embeds in a rational homology 3-ball. Assume that A is equipped with two pseudo-parallelizations $\tau_0$ and $\tau_1$ that coincide with a common genuine parallelization along a regular neighborhood of $\partial A$. For $i \in \underline{3n}$, let $\omega_{0,i}(\tau_0)$ be a boundary form of $(A, \tau_0)$ and let $\omega_{1,i}(\tau_1)$ be a boundary form of $(A, \tau_1)$. There exists a closed 2-form $\omega(i)$ on $[0,1] \times UA$ that restricts

• to $\{0\} \times UA$ as $\omega_{0,i}(\tau_0)$

• to $\{1\} \times UA$ as $\omega_{1,i}(\tau_1)$,

• to $[0,1] \times UA|_{\partial A}$ as $(\mathbf{1}_{[0,1]} \times p_{\tau_0})^* (\omega_{S,i})$ with respect to a closed 2-form $\omega_{S,i}$ on $[0,1] \times S^2$ and to the projection $\mathbf{1}_{[0,1]} \times p_{\tau_0}: [0,1] \times UA|_{\partial A} \to [0,1] \times S^2$.

Let $n$ be a natural integer. As in Corollary 9.4, set

$$
z_n([0,1] \times UA; (\omega(i))_{i \in \underline{3n}}) = \sum_{\Gamma \in \mathcal{D}_n^c} \zeta_{\Gamma} \int_{[0,1] \times \tilde{S}_{V(\Gamma)}(TA)} \bigwedge_{e \in E(\Gamma)} p_e^*(\omega(j_E(e))) [\Gamma].
$$

Then we have

$$
z_n([0,1] \times UA; (\omega(i))_{i \in \underline{3n}}) = \frac{p_1(\tau_0, \tau_1)}{4} \beta_n.
$$

PROOF: The existence of $\omega_{S,i}$ is proved in Lemma 9.1. The proof of the existence of the form $\omega(i)$ with its prescribed properties is obtained from the proof of Proposition 19.23 by replacing Proposition 19.22 with Lemma 21.5.

Let $t$ stand for the coordinate in $[0,1]$. Assume that $\omega_{0,i}(\tau_0)$ is a homoge-
neous boundary form and that we have $\tau_0 = \tau_1$. Choose $\omega_{S,i} = \omega_{S^2} + d(t\eta_{i,s,1})$
with the notation before Lemma 21.1. Then the form $\omega(i)$ can be chosen so
that it is equal to

$$
p(\tau_e)^* (\omega_{S,i}) \text{ on } [0,1] \times U \left( A \setminus \left( [a, b - \epsilon_i + \epsilon_i^k] \times \gamma \times [-1, 1] \right) \right),
$$

$$
p(\tau_b)^*(\omega_{S^2}) + d(t\eta_{i,s}) \text{ on } [0,1] \times U([b - \epsilon_i - \epsilon_i^k, b - \epsilon_i + \epsilon_i^k] \times \gamma \times [-1, 1]), \text{ and}
$$

$$
\frac{1}{2} \left( p(\tau_b \circ \mathcal{T}_{\gamma}^{-1})^* (\omega_{S,i}) + p(\tau_b \circ F(\gamma, \tau_b)^{-1})^* (\omega_{S,i}) \right)
$$
on $[0, 1] \times U([a, b - \epsilon_i - \epsilon_i^k] \times \gamma \times [-1, 1])$. In this case, the parts over $(A \setminus ([a, b] \times \gamma \times [-1, 1]))$ in $z_n([0, 1] \times UA; (\omega(i))_{i \in \overline{3n}})$ cancel because the forms do not depend on the factor $A$ (which is factored out via the parallelization $\tau_e$), and the parts over $([a, b] \times \gamma \times [-1, 1])$ also cancel because the forms do not depend on the factor $\gamma$. So we simply find $z_n([0, 1] \times UA; (\omega(i))_{i \in \overline{3n}}) = 0$ as announced.

In general, as in the beginning of the proof of Proposition 19.24, and as in Corollary 9.4, the sum $z_n([0, 1] \times UA; (\omega(i)))$ depends only on the restriction of the $\omega(i)$ to $\partial([0, 1] \times UA)$. The above arguments can also be used to prove that $z_n([0, 1] \times UA; (\omega(i))_{i \in \overline{3n}})$ is independent of the forms $\omega_{S,i}$. So $z_n([0, 1] \times UA; (\omega(i))_{i \in \overline{3n}})$ depends only on the pairs $(\omega_{0,i}(\tau_0), \omega_{1,i}(\tau_1))$. Denote it by $z_n((\omega_{0,i}(\tau_0), \omega_{1,i}(\tau_1))_{i \in \overline{3n}})$. Observe

$$z_n((\omega_{0,i}(\tau_0), \omega_{2,i}(\tau_2))_{i \in \overline{3n}}) = z_n((\omega_{0,i}(\tau_0), \omega_{1,i}(\tau_1))_{i \in \overline{3n}}) + z_n((\omega_{1,i}(\tau_1), \omega_{2,i}(\tau_2))_{i \in \overline{3n}})$$

again. Conclude with the study of the special case and with Theorem 19.25. $\square$

<figure><img src="image_165.png" /></figure>

**Definition 21.7.** Let $\tilde{R}$ be a rational homology $\mathbb{R}^3$ equipped with an asymptotically standard pseudo-parallelization $\tilde{\tau}$. A propagating form of $(C_2(R), \tilde{\tau})$ is a propagating form of $C_2(R)$ (as in Definition 3.11) that coincides with a boundary form (of Definition 21.4) of $(\tilde{R}, \tilde{\tau})$ on $U\tilde{R}$.

**Lemma 21.8.** Let $\tilde{R}$ be a rational homology $\mathbb{R}^3$ equipped with an asymptotically standard pseudo-parallelization $\tau = (N(\gamma); \tau_e, \tau_b)$. Let $K: \mathcal{L} \hookrightarrow \tilde{R}\setminus N(\gamma)$ be a knot embedding, which is straight with respect to $\tau|_{\tilde{R}\setminus N(\gamma)}$, as in Definition 7.37. Let $K_{\parallel, \tau}$ denote the parallel of $K$ induced by $\tau$, as in Lemma 7.35. For any propagating form $\omega_p$ of $(C_2(R), \tau)$, we have

$$\int_{C(K; \xleftarrow{\phantom{\tau}} S^1)} \omega_p = I_{\theta}(K, \tau) = lk(K, K_{\parallel, \tau, Y}).$$

**PROOF:** Lemma 7.35 extends to this case. See also Lemmas 7.13 and 21.3. $\square$

<figure><img src="image_166.png" /></figure>

Theorem 7.40 generalizes as follows to pseudo-parallelizations:

**Theorem 21.9.** Let $\tilde{R}$ be a rational homology $\mathbb{R}^3$ equipped with an asymptotically standard pseudo-parallelization $\tau = (N(\gamma); \tau_e, \tau_b)$. Let $L: \mathcal{L} \hookrightarrow \tilde{R}\setminus N(\gamma)$ be a link embedding, which is straight with respect to $\tau|_{\tilde{R}\setminus N(\gamma)}$. Let $L_{\parallel, \tau}$ denote
the parallel of L induced by $\tau$. For any $i \in \underline{3n}$, let $\omega(i)$ be a propagating form of $(C_2(R), \tau)$. Then

$$Z_n^s(\check{R}, L, (\omega(i))_{i \in \underline{3n}}) = \sum_{\Gamma \in \mathcal{D}_n^e(\mathcal{L})} \zeta_{\Gamma} I(R, L, \Gamma, (\omega(i))_{i \in \underline{3n}}) [\Gamma] \in \mathcal{A}_n(\mathcal{L}).$$

is independent of the chosen $\omega(i)$. Set

$$Z_n^s(\check{R}, L, \tau) = Z_n^s(\check{R}, L, (\omega(i))_{i \in \underline{3n}})$$

and $Z^s(\check{R}, L, \tau) = (Z_n^s(\check{R}, L, \tau))_{n \in \mathbb{N}}$. Then we have

$$Z^s(\check{R}, L, \tau) = Z(\check{R}, L, \tau) = \mathcal{Z}^f(\check{R}, L, L_{\parallel, \tau}) \exp \left( \frac{p_1(\tau)}{4} \beta \right),$$

where $Z(\check{R}, L, \tau)$ is defined in Theorem 19.17 (as in Theorem 7.20), with Definition 19.12 for $p_1(\tau)$, and Definition 7.41 for $\mathcal{Z}^f(\check{R}, L, L_{\parallel, \tau})$.

Theorem 21.9 is a consequence of the following generalization of Theorem 16.9 to pseudo-parallelizations:

**Theorem 21.10.** Let $\mathcal{C}$ be a rational homology cylinder equipped with a pseudo-parallelization $\tau = (N(\gamma); \tau_e, \tau_b)$. Let $L: \mathcal{L} \hookrightarrow R(\mathcal{C}) \setminus N(\gamma)$ be a long tangle representative associated to a tangle in $\mathcal{C}$. Let $\{K_j\}_{j \in I}$ be the set of components of $L$. Assume that the bottom (resp. top) configuration of $L$ is represented by a map $y^-: B^- \to D_1$ (resp. $y^+: B^+ \to D_1$).

Let $N \in \mathbb{N}$. For $i \in \underline{3N}$, let $\tilde{\omega}(i, S^2) = (\tilde{\omega}(i, t, S^2))_{t \in [0,1]}$ be a closed 2-form on $[0, 1] \times S^2$ such that $\int_{S^2} \tilde{\omega}(i, 0, S^2) = 1$. There exists a closed 2-form $\tilde{\omega}(i) = (\tilde{\omega}(i, t))_{t \in [0,1]}$ on $[0, 1] \times C_2(R(\mathcal{C}))$ such that

$$\tilde{\omega}(i) = (\mathbf{1}_{[0,1]} \times p_{\tau_e})^* (\tilde{\omega}(i, S^2))$$

on $[0, 1] \times (\partial C_2(R(\mathcal{C})) \setminus UN(\gamma))$ and the restriction of $\tilde{\omega}(i, t)$ to $U\check{R}(\mathcal{C})$ is a boundary form of $(\check{R}(\mathcal{C}), \tau)$ as in Definition 21.4 for all $t \in [0, 1]$.

For such a family $(\tilde{\omega}(i))_{i \in \underline{3N}}$, and for a subset $A$ of $\underline{3N}$ with cardinality 3k, set

$$Z(C, L, \tau, A, (\tilde{\omega}(i, t))_{i \in A}) = \sum_{\Gamma \in \mathcal{D}_{\check{k}, A}(\mathcal{L})} \zeta_{\Gamma} I(C, L, \Gamma, (\tilde{\omega}(i, t))_{i \in A}) [\Gamma] \in \mathcal{A}_k(\mathcal{L}).$$

Then

$$Z(C, L, \tau, A)(t) = Z(C, L, \tau, A, (\tilde{\omega}(i, t))_{i \in A})$$
depends only on $(\tilde{\omega}(i, t, S^2))_{i \in A}$ for any $t$ (and on $(C, L, \tau)$).

It will be denoted by $Z(C, L, \tau, A, (\tilde{\omega}(i, t, S^2))_{i \in A})$. When $\tilde{\omega}(i, t, S^2)$ is the standard homogeneous form $\omega_{S^2}$ on $S^2$ for any $i$ and for all $t \in [0, 1]$, the map $Z(C, L, \tau, \cdot)(t)$ sends any subset of $3N$ with cardinality $3k$ to $Z_k(C, L, \tau)$.

Furthermore, with the notation of Definition 16.10, for any given orientation of $L$, we have

$$ Z(C, L, \tau, \cdot)(t) = \left( \left( \prod_{j \in I} \widetilde{\operatorname{hol}}_{[0,t]} (\eta(\cdot, U_j^+)) \right) \#_j \widetilde{\operatorname{hol}}_{[t,0] \times y^-} (\eta_{B^-, \cdot}) Z(C, L, \tau, \cdot) (0) \widetilde{\operatorname{hol}}_{[0,t] \times y^+} (\eta_{B^+, \cdot}) \right) \rfloor $$

where $U_j^+ = p_\tau(U^+ K_j)$.

**PROOF:** Fix $i \in 3N$. Pick a one-form $\tilde{\eta}_{i,s,1}$ on $[0, 1] \times S^2$ such that $\tilde{\omega}(i, S^2) = p_{S^2}^*(\omega_{S^2}) + d\tilde{\eta}_{i,s,1}$. Use this form to construct a one-form $\tilde{\eta}_{i,s}(t)$ on $[0, 1] \times U([a, b] \times \gamma \times [-1, 1])$ that pulls back through $p_{[0,1]} \times p_{[a,b]} \times p_{[-1,1]} \times p(\tau_b)$ and such that its restriction to $\{t\} \times U(N(\gamma))$ satisfies the conditions of Lemma 21.1, with respect to $\tilde{\eta}_{i,s,1}|_{\{t\} \times U(N(\gamma))}$.

Next use $\tilde{\eta}_{i,s}$ to construct the restriction of $\tilde{\omega}(i)$ to $[0, 1] \times \partial C_2(R(C))$, such that $\tilde{\omega}(i, t)|_{\{t\} \times U\tilde{R}(C)} = \omega(\tau, \tilde{\omega}(i, t, S^2), k, \varepsilon_i, \tilde{\eta}_{i,s}(t))$. The existence of $\tilde{\omega}(i)$ follows, as in Lemma 9.1.

Use the proof of Theorem 16.9 to prove the variation formula that expresses $Z(C, L, \tau, A, (\tilde{\omega}(i, t))_{i \in A})$ as a function of $Z(C, L, \tau, A, (\tilde{\omega}(i, 0))_{i \in A})$ for forms $\tilde{\omega}(i, t)$ as in the statement.

Then note that if $\tilde{\omega}(i, 0, S^2)$ is the standard homogeneous form $\omega_{S^2}$ on $S^2$ for any $i$, then $Z(C, L, \tau, \cdot)(0)$ maps any subset of $3N$ with cardinality $3k$ to $Z_k(C, L, \tau)$, by Theorem 19.17. Thanks to Lemma 9.1, this shows that $Z(C, L, \tau, A)(t)$ depends only on the forms $\tilde{\omega}(i, S^2)$, and therefore only on the forms $\tilde{\omega}(i, t, S^2)$. $\square$

**PROOF OF THEOREM 21.9:** Apply the formula of Theorem 21.10. There are no factors $\widetilde{\operatorname{hol}}_{[t,0] \times y^-} (\eta_{B^-, \cdot})$ and $\widetilde{\operatorname{hol}}_{[0,t] \times y^+} (\eta_{B^+, \cdot})$ because $L$ is a link. Since $L$ is a straight link with respect to $\tau$, the factors $\widetilde{\operatorname{hol}}_{[0,t]} (\eta(\cdot, p_\tau(U^+ K_j)))$ also vanish, even though the notion of straight link of Definition 7.37 is less restrictive than the notion of straight tangle of Definition 16.11. Theorem 21.9 follows from Theorems 21.10 and 19.17, and Lemma 21.8. $\square$

**Remark 21.11.** When $\tilde{\omega}(i, 1, S^2)$ is the standard homogeneous form $\omega_{S^2}$ on $S^2$ for any $i$, the variation formula of Theorem 21.10 yields alternative expressions of $Z$.
## 21.2 Pseudo-sections associated with pseudo-parallelizations

For an asymptotically standard parallelization $\tau$ of a punctured rational homology 3-sphere $\tilde{R}$, a propagating chain of $(C_2(R), \tau)$ is defined in Definition 3.9 to be a 4-chain $P$ of $C_2(R)$ such that $\partial P = p_{\tau}^{-1}(X)$ for some $X \in S^2$, where $p_{\tau}^{-1}(X) \subset \partial C_2(R)$, the intersection $p_{\tau}^{-1}(X) \cap (\partial C_2(R) \setminus U\tilde{R})$ is independent of $\tau$, and

$$p_{\tau}^{-1}(X) \cap U\tilde{R} = \tau(\tilde{R} \times \{X\}).$$

The image in $UA$ of the restriction of a section $\tau(\tilde{R} \times X) = \tau(\tilde{R} \times \{X\})$ to a part $A$ of $\tilde{R}$ is denoted by $s_{\tau}(A; X)$.

In this section, we define pseudo-sections $s_{\tilde{\tau}}(\tilde{R}; X)$ associated with pseudo-parallelizations $\tilde{\tau}$. A propagating chain of $(C_2(R), \tilde{\tau})$ is a 4-chain $P$ of $C_2(R)$ such that

$$\partial P = (p_{\tau}|_{\partial C_2(R) \setminus U\tilde{R}})^{-1}(X) \cup s_{\tilde{\tau}}(\tilde{R}; X)$$

for some $X \in S^2$. Thus, the pseudo-sections $s_{\tilde{\tau}}(\tilde{R}; X)$ will play the same role as the sections $s_{\tau}(\tilde{R}; X)$ in the more flexible definition of $\mathcal{Z}$ presented in Section 21.3.

**Definition 21.12** (Pseudo-sections $s_{\tilde{\tau}}(\tilde{R}; X)$). Recall the map $F(\gamma, \tau_b)$ of Definition 19.13 and the notation of Definition 19.9.

Let $X \in S^2$. Let $S^1(X)$ be the intersection of $S^2$ with the plane orthogonal to the axis generated by $(0, 0, 1)$ that contains $X$. So $S^1(X)$ is a circle or a point. Let $G_2(X)$ be a 2-dimensional chain in $[-1, 1] \times S^1(X)$ whose boundary is $\{(u, \rho_{-\alpha(u)}(X)), u \in [-1, 1]\} + \{(u, \rho_{\alpha(u)}(X)), u \in [-1, 1]\} - 2[-1, 1] \times \{X\}$, as in Figure 21.1.

<figure><img src="image_167.png" /><figcaption>Figure 21.1: The chain $G_2(X)$ in $[-1, 1] \times S^1(X)$</figcaption></figure>

Then $s_{\tilde{\tau}}(A; X)$ is the following 3-cycle of $(UA, UA|_{\partial A})$

$$s_{\tilde{\tau}}(A; X) = s_{\tau_e}(A \setminus \tilde{N}(\gamma); X) + \frac{1}{2} \left( s_{\tau_b \circ \mathcal{T}_{\gamma}^{-1}}(N(\gamma); X) + s_{\tau_b \circ F(\gamma, \tau_b)^{-1}}(N(\gamma); X) + \{b\} \times \gamma \times G_2(X) \right),$$

where $UA|_{\{b\} \times \gamma \times [-1, 1]}$ is identified with $\{b\} \times \gamma \times [-1, 1] \times S^2$ by $\tau_b$ (or in the same way by $\tau_e$).
We also introduce small deformations of these sections, associated with $\epsilon_i$ such that $0 \le \epsilon_i < \epsilon$ (with respect to the $\epsilon$ of Definition 19.9) as follows. Let $N(\gamma, \epsilon_i) = [a, b - \epsilon_i] \times \gamma \times [-1, 1] \times S^2$. Extend $\tau_e$ over $[b - \epsilon, b] \times \gamma \times [-1, 1]$ so that it coincides with $\tau_b$, there. Then $s_{\tilde{\tau}}(A; X, \epsilon_i)$ is the following 3-cycle of $(UA, UA|_{\partial A})$

$$s_{\tilde{\tau}}(A; X, \epsilon_i) = s_{\tau_e}(A \setminus \mathring{N}(\gamma, \epsilon_i); X) + \frac{1}{2} \left( s_{\tau_b \circ \mathcal{T}_{\gamma}^{-1}}(N(\gamma, \epsilon_i); X) + s_{\tau_b \circ F(\gamma, \tau_b)^{-1}}(N(\gamma, \epsilon_i); X) \right) + \frac{1}{2} \{b - \epsilon_i\} \times \gamma \times G_2(X).$$

When $\Sigma$ is a 2-chain that intersects $N(\gamma)$ along sections $N_c(\gamma) = [a, b] \times \{c\} \times [-1, 1]$ (which are oriented as meridian disks of $(-\gamma)$), set

$$s_{\tilde{\tau}}(\Sigma; X) = s_{\tilde{\tau}}(A; X) \cap UA|_{\Sigma}.$$

So $s_{\tilde{\tau}}(N_c(\gamma); X)$ equals

$$\frac{1}{2} \left( s_{\tau_b \circ \mathcal{T}_{\gamma}^{-1}}(N_c(\gamma); X) + s_{\tau_b \circ F(\gamma, \tau_b)^{-1}}(N_c(\gamma); X) - \{b\} \times \{c\} \times G_2(X) \right).$$

Note that $G_2(e_3)$ lies in $[-1, 1] \times \{e_3\}$. So $s_{\tau}(A; \pm e_3)$ is simply given by

$$s_{\tau}(A; \pm e_3) = s_{\tau_e}(A \setminus \mathring{N}(\gamma); \pm e_3) + \frac{1}{2} \left( s_{\tau_b \circ \mathcal{T}_{\gamma}^{-1}}(N(\gamma); \pm e_3) + s_{\tau_b \circ F(\gamma, \tau_b)^{-1}}(N(\gamma); \pm e_3) \right).$$

Below, we discuss common properties of homology classes of sections and pseudo-sections.

**Lemma 21.13.** Let $W \in S^2$. Let $\Phi$ be a map from the unit disk $D^2$ of $\mathbb{C}$ to $SO(3)$ such that $\Phi(\exp(i\beta))$ is the rotation $\rho_{2\beta,W}$ of angle $2\beta$ whose axis is directed by $W$ for $\beta \in [0, 2\pi]$. Then the map

$$\Phi_W = \Phi(\cdot)(W): \begin{array}{ccc} D^2 & \to & S^2 \\ z & \mapsto & \Phi(z)(W) \end{array}$$

sends $\partial D^2$ to $W$, and the degree of the induced map from $D^2/\partial D^2$ to $S^2$ is $(-1)$.

**PROOF:** First note that the above degree does not depend on $\Phi$ on the interior of $D^2$ since $\pi_2(SO(3)) = 0$. View the restriction of $\Phi_W$ to $\partial D^2$, as the path composition of the maps $(\beta \in [0, 2\pi] \mapsto \rho_{\beta,W})$ and the inverse of $(\beta \in [0, 2\pi] \mapsto \rho_{\beta,-W})$ (which is homotopic to the first map). Consider an arc $\xi$ of a great circle of $S^2$ from $-W$ to $W$. Then $\Phi$ can be regarded as the map
from $\xi \times [0, 2\pi]$ to $SO(3)$ that maps $(X, \beta)$ to $\rho_{\beta,X}$. So the only preimage of $-W$ under $\Phi_W$ is $(X_0, \pi)$, where $X_0 \in \xi$ and $X_0 \perp W$. The local degree is easily seen to be $(-1)$. $\square$

Lemma 19.20 generalizes as follows to pseudo-parallelizations, with the notation of Definition 19.18.

**Proposition 21.14.** *Recall $e_3 = (0, 0, 1) \in \mathbb{R}^3$. Let $\Sigma$ be a compact oriented surface immersed in a 3-manifold $M$ equipped with a pseudo-parallelization $\tau = (N(\gamma); \tau_e, \tau_b)$. Assume that $\tau$ is an actual parallelization around the boundary $\partial\Sigma$ of $\Sigma$ and that $\tau(\cdot \times e_3)$ is a positive normal to $\Sigma$ along $\partial\Sigma$. Let $s_+(\Sigma) \subset UM$ be the section of $UM|_{\Sigma}$ in $UM$ associated to the positive normal field $n$ to $\Sigma$, which coincides with $\tau(\partial\Sigma \times e_3)$ on $\partial\Sigma$. Then*

$$2(s_+(\Sigma) - s_\tau(\Sigma; e_3)) - \chi(\tau(\cdot \times e_2)|_{\partial\Sigma}; \Sigma)UM|_*$$

and

$$2(s_-(\Sigma) - s_\tau(\Sigma; -e_3)) + \chi(\tau(\cdot \times e_2)|_{\partial\Sigma}; \Sigma)UM|_*$$

*are null-homologous cycles in $UM|_{\Sigma}$.*

**PROOF:** Lemma 19.20 gives the result for actual parallelizations. Let us prove it for pseudo-parallelizations. Since $s_\tau(\Sigma; \pm e_3) = s_\tau(M; \pm e_3) \cap UM|_\Sigma$ is well-defined for any surface $\Sigma$, as soon as the above intersection is transverse, there is no loss of generality in assuming that $\Sigma$ meets $N(\gamma)$ along sections $N_c(\gamma)$, for some $c \in \gamma$.

Since the cycles of the statement behave well under gluing (or cutting) surfaces along curves that satisfy the boundary assumptions, and since these assumptions are easily satisfied after a homotopy as in Definition 19.21 (when the cutting process is concerned), it suffices to prove the lemma for a meridian disk $\Sigma$ of $\gamma$. For such a disk, $s_\tau(\Sigma; e_3)$ is the average of two genuine sections corresponding to trivializations $\tau_1$ and $\tau_2$. So, with the notation of Lemma 19.19, the chain

$$2(s_+(\Sigma) - s_\tau(\Sigma; e_3)) - \left( \frac{d(\tau_1(\cdot \times e_2), \partial\Sigma)}{2} + \frac{d(\tau_2(\cdot \times e_2), \partial\Sigma)}{2} + \chi(\Sigma) \right) UM|_*$$

is a null-homologous cycle. We conclude that the first cycle is null-homologous because the exterior trivialization $\tau_e$ satisfies

$$d(\tau_e(\cdot \times e_2), \partial\Sigma) = \frac{d(\tau_1(\cdot \times e_2), \partial\Sigma) + d(\tau_2(\cdot \times e_2), \partial\Sigma)}{2}.$$

The second cycle can be treated similarly. $\square$

The obvious property that genuine sections $s_\tau(\Sigma; X)$ and $s_\tau(\Sigma; Y)$ corresponding to distinct $X$ and $Y$ of $S^2$ are disjoint generalizes as follows for pseudo-sections.
**Lemma 21.15.** Let $\Sigma$ be an oriented surface embedded in a 3-manifold $A$ equipped with a pseudo-parallelization $\tilde{\tau} = (N(\gamma); \tau_e, \tau_b)$ such that $\Sigma$ intersects $N(\gamma)$ only along sections $N_{c_i}(\gamma) = [a, b] \times \{c_i\} \times [-1, 1]$ in the interior of $\Sigma$. If $Y \in S^2$ and if $Z \in S^2 \setminus S^1(Y)$, then the algebraic intersection $\langle s_{\tilde{\tau}}(\Sigma; Y), s_{\tilde{\tau}}(\Sigma; Z) \rangle_{UA|_{\Sigma}}$ of $s_{\tilde{\tau}}(\Sigma; Y)$ and $s_{\tilde{\tau}}(\Sigma; Z)$ in $UA|_{\Sigma}$ is zero.

**PROOF:** Recall that $S^1(X)$ is the intersection circle of $S^2$ with the plane orthogonal to the axis generated by $e_3 = (0, 0, 1)$ that contains $X$. Consider the contribution to $\langle s_{\tilde{\tau}}(\Sigma; Y), s_{\tilde{\tau}}(\Sigma; Z) \rangle_{UA|_{\Sigma}}$ of an intersection point $c$ of $\gamma$ with $\Sigma$. Such a contribution may be expressed as $\pm \frac{1}{4}(d_Y(Z) + d_Z(Y))$ with

$$d_Y(Z) = \langle s_{\tau_b \circ F(\gamma, \tau_b)^{-1}}(N_c(\gamma); Z), s_{\tau_b \circ \mathcal{T}_{\gamma}^{-1}}(N_c(\gamma); Y) \rangle$$

and $d_Z(Y) = \langle s_{\tau_b \circ F(\gamma, \tau_b)^{-1}}(N_c(\gamma); Y), s_{\tau_b \circ \mathcal{T}_{\gamma}^{-1}}(N_c(\gamma); Z) \rangle$. We prove that both intersection numbers $d_Y(Z)$ and $d_Z(Y)$ are $\pm 1$ and that their signs are opposite. Consider an arc $\xi$ of great circle from $e_3$ to $-e_3$. Identify $\xi$ with $[a, b]$ via an orientation-preserving diffeomorphism, so that the map $F$ of Definition 19.13 can be regarded as the map that maps $(V \in \xi, u \in [-1, 1])$ to the rotation $\rho_{\alpha(u),V}$ with axis $V$ and angle $\alpha(u)$. Then the intersection

$$d_Y(Z) = \langle s_{\tau_b \circ F(\gamma, \tau_b)^{-1}}(N_c(\gamma); Z), s_{\tau_b \circ \mathcal{T}_{\gamma}^{-1}}(N_c(\gamma); Y) \rangle$$

in

$$UA|_{N_c(\gamma)} = \tau_b \circ F(\gamma, \tau_b)^{-1} N_c(\gamma) \times S^2$$

is the algebraic intersection

$$\langle N_c(\gamma) \times \{Z\}, F(\gamma, \tau_b) \circ \mathcal{T}_{\gamma}^{-1}(N_c(\gamma) \times \{Y\}) \rangle_{N_c(\gamma) \times S^2}$$

It is the degree of the map

$$f_Y: \begin{array}{ccc} \xi \times [-1, 1] & \rightarrow & S^2 \\ (V, u) & \mapsto & \rho_{\alpha(u),V} \circ \rho_{\alpha(-u),e_3}(Y) \end{array}$$

at $Z$. The other algebraic intersection $d_Z(Y)$ is the degree of the map $f_Z$ at $Y$. The boundary of the image of $f_Y$ is $-2S^1(Y)$, where $S^1(Y)$ is oriented as the boundary of the closure of the connected component of $S^2 \setminus S^1(Y)$ that contains $e_3$. Therefore, the degree increases by 2 from the component of $S^2 \setminus S^1(Y)$ that contains $e_3$ to the component of $(-e_3)$. Furthermore, the degree of $f_Y$ on the component of $e_3$ is independent of $Y \neq e_3$. Lemma 21.13 implies $\deg(f_{-e_3}) = \deg(\Phi_{-e_3}) = -1$. So the degree of $f_Y$ at $e_3$ is $-1$ when $Y \neq e_3$. Therefore, the degree of $f_Y$ on the component of $-e_3$, which is independent of $Y \neq (-e_3)$, is 1. Thus, the degree of $f_Y$ at $Z$ and the degree of $f_Z$ at $Y$ are opposite. $\square$