**Lemma 21.16.** Let A be a rational homology handlebody equipped with two pseudo-parallelizations $\tilde{\tau}_0$ and $\tilde{\tau}_1$ that coincide with the same genuine parallelization near the boundary of A. Let $X \in S^2$. Let $\eta \in [0, \varepsilon]$. There exists a rational 4-chain $H(\tilde{\tau}_0, \tilde{\tau}_1, X, \eta)$ in UA such that

$$ \partial H(\tilde{\tau}_0, \tilde{\tau}_1, X, \eta) = s_{\tilde{\tau}_1}(A; X, \eta) - s_{\tilde{\tau}_0}(A; X, \eta). $$

**PROOF:** Let $Y \in S^2 \setminus S^1(X)$. Let us show that $C = s_{\tilde{\tau}_1}(A; X, \eta) - s_{\tilde{\tau}_0}(A; X, \eta)$ vanishes in

$$ H_3(UA; \mathbb{Q}) = H_1(A; \mathbb{Q}) \otimes H_2(S^2; \mathbb{Q}). $$

To do so, we prove that the algebraic intersections of $C$ with $s_{\tilde{\tau}_0}(S; Y)$ vanish for 2-cycles $S$ of $(A, \partial A)$ that generate $H_2(A, \partial A)$. We assume $\eta = 0$ without loss. Of course $s_{\tilde{\tau}_0}(\partial A; X)$ does not intersect $s_{\tilde{\tau}_0}(S; Y)$. According to Lemma 21.15, the chain $s_{\tilde{\tau}_0}(S; X)$ does not intersect $s_{\tilde{\tau}_0}(S; Y)$ in $UA|_S$ algebraically. So $s_{\tilde{\tau}_0}(A; X)$ does not intersect $s_{\tilde{\tau}_0}(S; Y)$ algebraically, either. Since Proposition 21.14 guarantees that $(s_{\tilde{\tau}_0}(S; X) - s_{\tilde{\tau}_1}(S; X))$ bounds in $UA|_S$, the chain $s_{\tilde{\tau}_1}(S; X)$ does not intersect $s_{\tilde{\tau}_0}(S; Y)$ algebraically in $UA|_S$. So $s_{\tilde{\tau}_1}(A; X)$ does not intersect $s_{\tilde{\tau}_0}(S; Y)$ algebraically. This proves the existence of the desired 4-chain $H(\tilde{\tau}_0, \tilde{\tau}_1, X, \eta)$. $\square$

## 21.3 Definition of $\mathcal{Z}$ with respect to pseudo-sections

**Definition 21.17.** Let $\tilde{R}$ be a rational homology $\mathbb{R}^3$ equipped with an asymptotically standard pseudo-parallelization $\tilde{\tau}$. A propagating chain of $(C_2(R), \tilde{\tau})$ is a propagating chain of $C_2(R)$ (as in Definition 3.11) whose boundary intersects $U\tilde{R}$ as a chain $s_{\tilde{\tau}}(\tilde{R}; X, \varepsilon_i)$ as in Definition 21.12.

Again, for a given $s_{\tilde{\tau}}(\tilde{R}; X, \varepsilon_i)$, a propagating chain whose boundary intersects $U\tilde{R}$ as $s_{\tilde{\tau}}(\tilde{R}; X, \varepsilon_i)$ exists because $H_3(C_2(R); \mathbb{Q}) = 0$.

**Lemma 21.18.** Let $\tilde{R}$ be a rational homology $\mathbb{R}^3$ equipped with an asymptotically standard pseudo-parallelization $\tilde{\tau}$. For any positive number $\alpha$, and for any propagating chain $P$ of $(C_2(R), \tilde{\tau})$, transverse to $\partial C_2(R)$, there exists a propagating form of $(C_2(R), \tilde{\tau})$ (as in Definition 21.7) that is $\alpha$-dual to $P$ (as in Definition 11.6).

**PROOF:** There is no loss of generality in assuming that $P$ intersects a collar $[-1, 0] \times \partial C_2(R)$ as $[-1, 0] \times \partial P$.
Assume that the boundary of P intersects $U\tilde{R}$ as a chain $s_{\tilde{\tau}}(\tilde{R}; X, \varepsilon_i)$. Let $\omega_i$ be volume-one form of $S^2$, which is $\alpha_X$-dual to $X$ for some small $\alpha_X \in ]0, \alpha[$. Set

$$I(\varepsilon_i) = [b - \varepsilon_i - \varepsilon_i^k, b - \varepsilon_i + \varepsilon_i^k]$$

and $I_2(\varepsilon_i) = [b - \varepsilon_i - 2\varepsilon_i^k, b - \varepsilon_i + 2\varepsilon_i^k] \times [-1, 1] \times S^2$.

Forms $\alpha$-dual to a given chain can be constructed as in Lemma B.4. We use this process to construct a form $\omega_1$ $\alpha$-dual to P, which pulls back through $\partial C_2(R)$ on $[-1, 0] \times \partial C_2(R)$, which may be expressed as in Definition 21.2 outside $[-1, 0] \times U(I(\varepsilon_i) \times \gamma \times [-1, 1])$, and which factors through $I(\varepsilon_i) \times [-1, 1] \times S^2$ on $[-1, 0] \times U(I(\varepsilon_i) \times \gamma \times [-1, 1])$.

It remains to see that $\omega_1$ can be assumed to have the form prescribed by Definition 21.7 on $[-1, 0] \times U(I(\varepsilon_i) \times \gamma \times [-1, 1])$.

We also have a form $\omega_2$ as in Definition 21.7, which also pulls back through $\partial C_2(R)$ on $[-1, 0] \times \partial C_2(R)$ and which coincides with $\omega_1$ on $[-1, 0] \times U(\tilde{R} \setminus (I(\varepsilon_i) \times \gamma \times [-1 + \varepsilon, 1 - \varepsilon]))$.

Both forms factor through $I_2(\varepsilon_i) \times [-1, 1] \times S^2$, and they are cohomologous there. So there exists a one-form $\eta$ on $I_2(\varepsilon_i) \times [-1, 1] \times S^2$ such that $\omega_2 - \omega_1 = p^*(d\eta)$, where $p$ is the natural projection that forgets the $[-1, 0]$ factor and the $\gamma$ factor, and $d\eta$ vanishes outside $I(\varepsilon_i) \times [-1 + \varepsilon, 1 - \varepsilon] \times S^2$.

To conclude the proof, it suffices to prove that such $\eta$ can be chosen so that $\eta$ is zero outside $I(\varepsilon_i) \times [-1 + \varepsilon, 1 - \varepsilon] \times S^2$, too. Therefore, it suffices to prove that the cohomology class of $\eta$ is zero on

$$((I_2(\varepsilon_i) \times [-1, 1]) \setminus (I(\varepsilon_i) \times [-1 + \varepsilon, 1 - \varepsilon])) \times S^2.$$

Thus, it suffices to prove that the integral of $\eta$ along

$$\partial(I_2(\varepsilon_i) \times [-1, 1] \times \{Y\})$$

is zero for some $Y \in S^2$. This integral is also the integral of $(\omega_2 - \omega_1)$ along $I_2(\varepsilon_i) \times [-1, 1] \times \{Y\}$ or along $(N_c(\gamma) = [a, b] \times \{c\} \times [-1, 1]) \times \{Y\}$, where the integral of $\omega_1$ is the algebraic intersection of $s_{\tau_b}([a, b] \times \{c\} \times [-1, 1]; Y)$ with P.

Note that nothing depends on the trivialization $\tau_b$, which identifies

$$U([a, b] \times \gamma \times [-1, 1])$$

with $([a, b] \times \gamma \times [-1, 1]) \times S^2$. Therefore, there is no loss of generality in assuming that $\tau_b$ maps $e_3$ to the direction of $(-\gamma)$, which is the positive normal to the meridian disks, so that $(N_c(\gamma) \times \{e_3\}) = s_+(N_c(\gamma))$.
Assume $X \neq e_3$ and choose $Y = e_3$. Then it suffices to prove $\int_{s_+(N_c(\gamma))} \omega_1 = \int_{s_+(N_c(\gamma))} \omega_2$. Lemma 21.5 implies

$$ \int_{s_+(N_c(\gamma))} \omega_2 = \frac{1}{2} \chi(\tau_e(\cdot \times e_2)|_{\partial N_c(\gamma)}; N_c(\gamma)). $$

Let us compute $\int_{s_+(N_c(\gamma))} \omega_1 = \langle s_+(N_c(\gamma)), P \rangle$. Thanks to Proposition 21.14,

$$ 2(s_+(N_c(\gamma)) - s_{\tilde{\tau}}(N_c(\gamma); e_3)) - \chi(\tau_e(\cdot \times e_2)|_{\partial N_c(\gamma)}; N_c(\gamma)) U\tilde{R}|_* $$

is null-homologous in $U\tilde{R}|_*$. So its algebraic intersection with $P$ is zero. The algebraic intersection of $s_{\tilde{\tau}}(N_c(\gamma); e_3)$ with $P$ is also zero according to Lemma 21.15. We thus get $\int_{s_+(N_c(\gamma))} \omega_1 = \int_{s_+(N_c(\gamma))} \omega_2$, and the lemma is proved when $e_3 \neq X$.

When $e_3 = X$, choose $Y = -e_3$ and conclude by computing the integrals along $s_-(N_c(\gamma))$, similarly. $\square$

For a family of propagating chains $P_i$ of $(C_2(R), \tilde{\tau})$ in general $3n$-position with respect to $L$ as in Definition 11.3, and for propagating forms $\omega(i)$, which are $\alpha$-dual to them for a sufficiently small $\alpha$ as in Lemma 21.18, the integrals involved in $Z_n^s(\tilde{R}, L, \tilde{\tau})$ in Theorem 21.9 can be computed as algebraic intersections of preimages of transverse propagating chains of $(C_2(R), \tilde{\tau})$ as in Sections 11.1 and 17.1.

This provides the announced discrete definition of $Z_n^s(\tilde{R}, L, \tilde{\tau})$ with respect to pseudo-parallelizations.
# Appendix A

## Some basic algebraic topology

This appendix reviews the main well-known results in algebraic topology used throughout the book. The proofs of these results can be found in several books about basic algebraic topology, such as [Gre67, Hat02, Spa81], by Marvin Greenberg, Allen Hatcher, and Edwin Spanier. Here we state only the weak versions used in the book, with some sketches of proofs and some hints to show how things work.

### A.1 Homology

We first review some properties of homology.

A topological pair $(X, Y)$ consists of a topological space $X$ and a subspace $Y$ of $X$. A map $f: (X, Y) \to (A, B)$ between such pairs is a continuous map $f$ from $X$ to $A$ sending $Y$ to $B$. A topological space $X$ is identified with the pair $(X, \emptyset)$.

The coefficient ring $\Lambda$ of the homology theories $H(\cdot) = H(\cdot; \Lambda)$ considered in this book is $\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}$, $\mathbb{Q}$, or $\mathbb{R}$.

A covariant functor $H$ from the category of topological pairs $(X, Y)$ to the category of graded $\Lambda$-modules and homomorphisms of degree 0 maps any topological pair $(X, Y)$ to a sequence $H(X, Y; \Lambda) = H(X, Y) = (H_k(X, Y))_{k \in \mathbb{Z}}$ of $\Lambda$-modules. It associates to any map $f: (X, Y) \to (A, B)$ between pairs a sequence of $\Lambda$-linear morphisms $H(f) = (H_k(f): H_k(X, Y) \to H_k(A, B))_{k \in \mathbb{Z}}$ so that if $g: (A, B) \to (V, W)$ is another map between pairs $H_k(g \circ f) = H_k(g) \circ H_k(f)$, and $H_k$ maps the identity map of $(X, Y)$ to the identity map of $H_k(X, Y)$.

The homology theories $(H, \partial)$ that we consider consist of

* a covariant functor $H$ from the category of topological pairs $(X, Y)$ to the category of graded $\Lambda$-modules and homomorphisms of degree 0,
and

• sequences $\partial_k(X, Y)$ of linear maps $\partial_k(X, Y): H_k(X, Y) \to H_{k-1}(Y)$ associated to topological pairs $(X, Y)$ such that for any map $f: (X, Y) \to (A, B)$, we have

$$ \partial_k(A, B) \circ H_k(f) = H_{k-1}(f|_Y) \circ \partial_k(X, Y). $$

The considered homology theories satisfy the following Eilenberg–Steenrod axioms [Spa81, Chap. 4, Sec. 8, p.199].

• **Homotopy axiom:** For two topological pairs $(X, Y)$ and $(A, B)$, if

$$ f: \begin{array}{ccc} [0, 1] \times X & \to & A \\ (t, x) & \mapsto & f_t(x) = f(t, x) \end{array} $$

is a continuous map sending $[0, 1] \times Y$ to $B$, then $H(f_0)$ is equal to $H(f_1)$.

• **Exactness axiom:** For any topological pair $(X, Y)$ with inclusion maps $i: Y \hookrightarrow X$ and $j: (X, \emptyset) \hookrightarrow (X, Y)$, there is a long exact sequence:

$$ \dots \xrightarrow{\partial_{k+1}(X,Y)} H_k(Y) \xrightarrow{H_k(i)} H_k(X) \xrightarrow{H_k(j)} H_k(X,Y) \xrightarrow{\partial_k(X,Y)} H_{k-1}(Y) \xrightarrow{H_{k-1}(i)} \dots \quad 1 $$

• **Excision axiom:** For any topological pair $(X, Y)$, if $U$ is an open subset of $X$ whose closure lies in the interior of $Y$, then the inclusion map $e: (X \setminus U, Y \setminus U) \to (X, Y)$ induces isomorphisms $H_k(e): H_k(X \setminus U, Y \setminus U) \to H_k(X, Y)$ for any $k \in \mathbb{Z}$.

• **Dimension axiom:** If $X$ has one element $x$, then $H_k(X) \cong \{0\}$ for any $k \neq 0$. Furthermore, $H_0(X)$ is isomorphic to $\Lambda$ and is equipped with a canonical generator denoted by $[x]$. So we write $H_0(\{x\}) = \Lambda[x]$.

An example of such a homology theory is the singular homology described in [Gre67, Chap. 10 and 13] and in [Hat02, Section 2.1].

The spaces considered in this book are homotopy equivalent to finite CW-complexes. The above Eilenberg–Steenrod axioms characterize the homology of these spaces. Let us show a few examples of properties of homology and computations from these axioms. First note that the functoriality implies that $H$ maps a homeomorphism $f: (X, Y) \to (A, B)$ between topological

<footnote n="1">¹Such a sequence of homomorphisms is exact if and only if the image of a homomorphism is equal to the kernel of the following one.</footnote>
pairs to an isomorphism $H(f)$. Also note that the homotopy axiom implies that the homologies of $\mathbb{R}^n$ and its unit ball $B^n$ are isomorphic to the homology of a point, which is determined by the dimension axiom.

Here are other easy consequences of the Eilenberg–Steenrod axioms.

**Proposition A.1.** Let $X$ and $Y$ be two topological spaces. Let $i_X$ and $i_Y$ denote their respective inclusion maps into their disjoint union $X \sqcup Y$. Then for any $k \in \mathbb{Z}$, $H_k(i_X): H_k(X) \to H_k(X \sqcup Y)$ and $H_k(i_Y): H_k(Y) \to H_k(X \sqcup Y)$ are injective, and we have

$$H_k(X \sqcup Y) = H_k(i_X)(H_k(X)) \oplus H_k(i_Y)(H_k(Y)).$$

**PROOF:** The excision isomorphism $H_k(e_X): H_k(X) = H_k(X, \emptyset) \to H_k(X \sqcup Y, Y)$ may be expressed as

$$H_k(e_X) = H_k(j: (X \sqcup Y, \emptyset) \hookrightarrow (X \sqcup Y, Y)) \circ H_k(i_X: X \hookrightarrow X \sqcup Y).$$

So $H_k(j)$ is surjective and $H_k(i_X)$ is injective. Similarly, $H_k(i_Y)$ is injective. Thus, the long exact sequence of $(X \sqcup Y, Y)$ yields short exact sequences

$$0 \to H_k(Y) \xrightarrow{H_k(i_Y)} H_k(X \sqcup Y) \xrightarrow{H_k(e_X)^{-1} \circ H_k(j)} H_k(X) \to 0$$

for any integer $k$. Since $(H_k(e_X)^{-1} \circ H_k(j)) \circ H_k(i_X)$ is the identity map, the exact sequence splits, and we get the result. □

**Proposition A.2.** Let $X$ be a topological space. Let $Y$ and $Z$ be two subspaces of $X$ such that $Z \subseteq Y$. Let

$$(\partial_k = \partial_k(X, Y, Z)): H_k(X, Y) \to H_{k-1}(Y, Z)$$

be the composition of $\partial_k(X, Y): H_k(X, Y) \to H_{k-1}(Y)$ and the map from $H_{k-1}(Y)$ to $H_{k-1}(Y, Z)$ induced by the inclusion. Then the long sequence

$$H_{k+1}(X, Y) \xrightarrow{\partial_{k+1}} H_k(Y, Z) \xrightarrow{H_k(i)} H_k(X, Z) \xrightarrow{H_k(j)} H_k(X, Y) \\ H_{k-1}(Y, Z) \xrightarrow{\partial_k} H_{k-1}(X, Z) \xrightarrow{H_{k-1}(j)} H_{k-1}(X, Y) \xrightarrow{\partial_{k-1}} H_{k-2}(Y)$$

where the $H_k(i)$ and the $H_k(j)$ are induced by inclusion, is exact. It is called the long exact sequence of homology of the triple $(X, Y, Z)$.
SKETCH OF PROOF: To prove that $H_k(j) \circ H_k(i) = 0$, note that the inclusion map from $(Y, Z)$ to $(X, Y)$ factors through $(Y, Y)$. Next chase in commutative diagrams using functoriality and the exactness axiom, which is this exact sequence when $Z = \emptyset$. $\square$

Let $X$ be a topological space with $k$ connected components. Choose a basepoint $x_i$ in each connected component $X_i$. Let $Y$ denote the set of these basepoints equipped with the natural discrete topology. Let $i: Y \hookrightarrow X$ be the inclusion map, and let $p$ be the map sending each connected component to its basepoint. By functoriality $H(p) \circ H(i)$ is the identity map. So $H(i)$ is injective. In particular $H_0(i)$ injects $\oplus_{x_i \in Y} \Lambda[x_i]$ into $H_0(X)$. The element $H_0(i)([x_i])$ is also denoted by $[x_i]$. The homotopy axiom implies that the element $[x_i]$ of $H_0(X_i)$ is independent of the basepoint $x_i$ of $X_i$ if $X_i$ is path-connected.

**Lemma A.3.** We have

$$ H_k(\mathbb{R}, \mathbb{R} \setminus \{0\}) = \left\{ \begin{array}{ll} \Lambda \partial_1^{-1}(\mathbb{R}, \mathbb{R} \setminus \{0\}) ([1] - [-1]) & \cong \Lambda \quad \text{if } k = 1 \\ \{0\} & \text{otherwise.} \end{array} \right. $$

**PROOF:** Proposition A.1 and the observations before imply

$$ H_0(\mathbb{R} \setminus \{0\}) = \Lambda[-1] \oplus \Lambda[+1]. $$

The morphism $H_0(i)$ induced by inclusion from $H_0(\mathbb{R} \setminus \{0\})$ to $H_0(\mathbb{R}) = \Lambda[1]$ maps $[-1]$ and $[+1]$ to the preferred generator $[-1] = [0] = [1]$ of $H_0(\mathbb{R})$. The long exact sequence associated to $(\mathbb{R}, \mathbb{R} \setminus \{0\}, \emptyset)$ implies that $\partial_1(\mathbb{R}, \mathbb{R} \setminus \{0\}): H_1(\mathbb{R}, \mathbb{R} \setminus \{0\}) \to H_0(\mathbb{R} \setminus \{0\})$ is an isomorphism onto its image, which is the kernel $\Lambda([+1] - [-1])$ of $H_0(i)$. The long exact sequence also implies $H_q(\mathbb{R}, \mathbb{R} \setminus \{0\}) = \{0\}$ when $q \neq 1$. $\square$

The reader can also apply the Eilenberg–Steenrod axioms and the above observations to prove the following proposition.

**Proposition A.4.** Let $n \in \mathbb{N} \setminus \{0\}$ and let $k \in \mathbb{Z}$. Recall

$$ B^n = \{ x = (x_1, \dots, x_n) \in \mathbb{R}^n : \|x\|^2 = \sum_{i=1}^n x_i^2 \le 1 \} $$

and $S^{n-1} = \partial B^n$.

Assume that $n \ge 2$. Set $B_+^{n-1} = \{x \in S^{n-1} : x_1 \ge 0\}$, $B_-^{n-1} = \{x \in S^{n-1} : x_1 \le 0\}$, and $S_+^{n-2} = \{x \in S^{n-1} : x_1 = 0\}$. The morphisms of the sequence

$$ H_k(\mathbb{R}^n, \mathbb{R}^n \setminus \{0\}) \xleftarrow{H_k(i)} H_k(B^n, S^{n-1}) \xrightarrow{\partial} H_{k-1}(S^{n-1}, B_-^{n-1}) \\ \leftarrow H_{k-1}(B_+^{n-1}, S_+^{n-2}) \xrightarrow{H_{k-1}(p)} H_{k-1}(\mathbb{R}^{n-1}, \mathbb{R}^{n-1} \setminus \{0\}), $$
where

• the unlabeled morphism and $H_k(i)$ are induced by inclusions,

• $\partial$ is the morphism $\partial_k(B^n, S^{n-1}, B_{-}^{n-1})$ of Proposition A.2, and

• $p: B_{+}^{n-1} \to \mathbb{R}^{n-1}$ forgets the first coordinate and shifts the numbering of the remaining ones by $(-1)$,

are isomorphisms. In particular, when $k = n$, their composition

$$D_n: H_n(\mathbb{R}^n, \mathbb{R}^n \setminus \{0\}) \to H_{n-1}(\mathbb{R}^{n-1}, \mathbb{R}^{n-1} \setminus \{0\})$$

is an isomorphism. Let $[\mathbb{R}, \mathbb{R} \setminus \{0\}] = \partial_1^{-1}(\mathbb{R}, \mathbb{R} \setminus \{0\})([1] - [-1])$ be the preferred generator of $H_1(\mathbb{R}, \mathbb{R} \setminus \{0\})$ of Lemma A.3. Inductively define

$$[\mathbb{R}^n, \mathbb{R}^n \setminus \{0\}] = D_n^{-1}[\mathbb{R}^{n-1}, \mathbb{R}^{n-1} \setminus \{0\}].$$

Then we have

$$H_k(\mathbb{R}^n, \mathbb{R}^n \setminus \{0\}) = \begin{cases} \Lambda[\mathbb{R}^n, \mathbb{R}^n \setminus \{0\}] \cong \Lambda & \text{if } k = n \\ \{0\} & \text{otherwise.} \end{cases}$$

for any positive integer $n$. For $n \ge 1$, set

$$[B^n, S^{n-1}] = H_n(i)^{-1}([\mathbb{R}^n, \mathbb{R}^n \setminus \{0\}]).$$

Then we have

$$H_k(B^n, S^{n-1}) = \begin{cases} \Lambda[B^n, S^{n-1}] \cong \Lambda & \text{if } k = n \\ \{0\} & \text{otherwise.} \end{cases}$$

□

As a corollary, we get the following proposition:

**Proposition A.5.** Let $n \in \mathbb{N}$. Assume that $n \ge 1$. Set

$$[S^n] = \partial_{n+1}(B^{n+1}, S^n) ([B^{n+1}, S^n]).$$

We have

$$H_k(S^n) = \begin{cases} \Lambda[S^n] \cong \Lambda & \text{if } k = n \\ \Lambda[(1, 0, \dots, 0)] \cong \Lambda & \text{if } k = 0 \\ \{0\} & \text{otherwise.} \end{cases}$$

**PROOF:** Use the exact sequence associated to $(B^{n+1}, S^n)$ and the previous proposition. □
**Proposition A.6.** Let $n \in \mathbb{N}$. Let $\phi$ be a diffeomorphism from $\mathbb{R}^n$ to $\mathbb{R}^n$ sending $0$ to $\phi(0) = 0$. If $\phi$ preserves the orientation, then $\phi$ induces the identity map on $H_n(\mathbb{R}^n, \mathbb{R}^n \setminus \{0\})$. Otherwise, we have

$$H_n(\phi)([\mathbb{R}^n, \mathbb{R}^n \setminus \{0\}]) = -[\mathbb{R}^n, \mathbb{R}^n \setminus \{0\}].$$

PROOF: When $n = 1$, this is easy to see with the generator

$$[\mathbb{R}, \mathbb{R} \setminus \{0\}] = \partial_1(\mathbb{R}, \mathbb{R} \setminus \{0\})^{-1}([+1] - [-1]).$$

Let $\iota_n: \mathbb{R}^n \to \mathbb{R}^n$ map $(x_1, x_2, \dots, x_{n-1}, x_n)$ to $(x_1, x_2, \dots, x_{n-1}, -x_n)$. With the notation of Proposition A.4, we have $H_{n-1}(\iota_{n-1}) \circ D_n = D_n \circ H_n(\iota_n)$. We inductively deduce $H_n(\iota_n)([\mathbb{R}^n, \mathbb{R}^n \setminus \{0\}]) = -[\mathbb{R}^n, \mathbb{R}^n \setminus \{0\}]$. Any orientation-reversing linear isomorphism of $\mathbb{R}^n$ is homotopic to $\iota_n$ through linear isomorphisms. Therefore, we have $H_n(\phi)([\mathbb{R}^n, \mathbb{R}^n \setminus \{0\}]) = -[\mathbb{R}^n, \mathbb{R}^n \setminus \{0\}]$ for any such isomorphism $\phi$. We similarly observe that $H_n(\phi)$ sends $[\mathbb{R}^n, \mathbb{R}^n \setminus \{0\}]$ to itself for an orientation-preserving linear isomorphism $\phi$. Diffeomorphisms preserving $0$ are homotopic to their linear derivative at $0$ through maps preserving $\mathbb{R}^n \setminus \{0\}$, near $0$. So the result follows for general diffeomorphisms thanks to the excision axiom. $\square$

Let $X$ be a topological space equipped with a triangulation $T$ as in Subsection 2.1.2. The Eilenberg–Steenrod axioms also imply that the homology $H(X)$ of $X$ can be computed as follows.<fnref n="2" /> Equip each simplex of the triangulation with an arbitrary orientation, and let $C_k(T) = C_k(T; \Lambda)$ denote the $\Lambda$-module freely generated by the simplices of dimension $k$ of $T$. The algebraic boundary of such a $k$-dimensional simplex $\Delta$ is the sum

$$\partial_k(\Delta) = \sum_{\delta} \varepsilon(\Delta, \delta)\delta,$$

running over all the $(k-1)$-dimensional simplices $\delta$ in the boundary of $\Delta$, where $\varepsilon(\Delta, \delta)$ is 1 if $\delta$ is oriented as (part of) the boundary of $\Delta$ (with the outward normal first convention as usual), and $(-1)$ otherwise. Define the *boundary map* $\partial_k: C_k(T) \to C_{k-1}(T)$ to be the linear map sending a $k$-dimensional simplex $\Delta$ to its algebraic boundary $\partial_k(\Delta)$. Then we have $\partial_k \circ \partial_{k+1} = 0$. So $(C(T), \partial) = (C_k(T), \partial)_{k \in \mathbb{Z}}$ is a *chain complex*. Its homology

$$H(C(T), \partial) = \left( H_k(C(T), \partial) = \frac{\text{Ker } \partial_k}{\text{Im } \partial_{k+1}} \right)_{k \in \mathbb{Z}}$$

is canonically isomorphic to the homology of $X$.

<footnote n="2">²The reader can prove it, but it takes more space.</footnote>
The elements of $\text{Ker } \partial_k$ are the $k$-dimensional cycles of $C(T)$. The elements of $\text{Im } \partial_{k+1}$ are the $k$-dimensional boundaries of $C(T)$. The elements of $C_k(T) = C_k(T; \Lambda)$ are called the simplicial chains of $T$ of dimension $k$ with coefficients in $\Lambda$

Thus, the homology of an $n$-dimensional manifold $M$ that can be equipped with a triangulation $T$ vanishes in degrees higher than $n$ and in negative degrees. (The existence of a homology theory satisfying the axioms and Proposition A.4 imply that the notion of dimension is well-defined for topological manifolds.) Let $C$ be an $n$-dimensional cycle in such a connected manifold $M$, and let $\Delta_1$ and $\Delta_2$ be two $n$-simplices of $T$ intersecting along an $(n-1)$-simplex. Then the coefficients of $\Delta_1$ and $\Delta_2$ in $C$ must coincide if the orientations of $\Delta_1$ and $\Delta_2$ are consistent along $\Delta_1 \cap \Delta_2$. They must be opposite to each other if the orientations are not consistent.

In particular, when $\Lambda = \mathbb{Z}/2\mathbb{Z}$, if $M$ is connected, then the existence of a nonzero $n$-dimensional cycle implies that the boundary of $M$ is empty and that $M$ is compact. When $\Lambda = \mathbb{Z}$, $\mathbb{Q}$ or $\mathbb{R}$, if $M$ is connected, then the existence of a nonzero $n$-dimensional cycle furthermore implies that $M$ is orientable.

Assume that $M$ is a compact, $n$-dimensional, connected, triangulable, oriented manifold with empty boundary, and assume that the $n$-simplices of $T$ are equipped with the induced orientation. Then the sum of all these simplices is a cycle. Its homology class is called the *fundamental class* of $M$. It is denoted by $[M]$. (When $\Lambda = \mathbb{Z}/2\mathbb{Z}$, this definition does not require an orientation of $M$.) Let $B^n$ be an $n$-dimensional closed ball embedded in $M$ by an orientation-preserving embedding. Let $0$ be the center of this ball. Thanks to the excision axion, the inclusions induce isomorphisms from $H_n(B^n, B^n \setminus \{0\})$ to $H_n(\mathbb{R}^n, \mathbb{R}^n \setminus \{0\})$, and from $H_n(B^n, B^n \setminus \{0\})$ to $H_n(M, M \setminus \{0\})$. Let $[B^n, B^n \setminus \{0\}]$ denote the generator of $H_n(B^n, B^n \setminus \{0\})$ mapped to $[\mathbb{R}^n, \mathbb{R}^n \setminus \{0\}]$ by the first isomorphism and let $[M, M \setminus \{0\}]$ denote the image of $[B^n, B^n \setminus \{0\}]$ under the second isomorphism. Then the inclusion induces an isomorphism from $H_n(M)$ to $H_n(M, M \setminus \{0\})$, which maps the generator $[M]$ of $H_n(M)$ to the generator $[M, M \setminus \{0\}]$ of $H_n(M, M \setminus \{0\})$. This defines the generator $[M]$ of $H_n(M)$ independently of a triangulation.

Let $X$ be a topological space equipped with a triangulation $T$ as above. Let $Y$ be a closed subspace of $X$ that is a union of simplices of $T$. Let $T_Y$ be the corresponding triangulation of $Y$. Set $C_k(T, T_Y) = C_k(T)/C_k(T_Y)$ and define $\partial_k(T, T_Y): C_k(T, T_Y) \to C_{k-1}(T, T_Y)$ to be the map induced by the previous boundary map $\partial_k$. Then the homology $H(X, Y)$ is canonically isomorphic to the homology of the chain complex $(C(T, T_Y), \partial(T, T_Y))$.

When $M$ is a connected, compact, oriented $n$-dimensional manifold with boundary, equipped with a triangulation $T$ whose $n$-dimensional simplices
are oriented by the orientation of $M$, the sum of all $n$-dimensional simplices of $T$ is a cycle of $C_n(M, \partial M)$. Its homology class is called the fundamental class of $(M, \partial M)$. It is denoted by $[M, \partial M]$. Again, if $B^n$ is an $n$-dimensional closed ball embedded in $M$ by an orientation-preserving embedding, then the inclusions induce isomorphisms from $H_n(B^n, B^n \setminus \{0\})$ to $H_n(M, M \setminus \{0\})$ and from $H_n(M, \partial M)$ to $H_n(M, M \setminus \{0\})$, and the image of $[B^n, B^n \setminus \{0\}]$ under the first isomorphism coincides with the image of $[M, \partial M]$ under the second one. Is is denoted by $[M, M \setminus \{0\}]$.

These considerations allow us to talk about the homology class of a compact oriented $p$-dimensional submanifold $P$ of a manifold $M$. It is the image of $([P] \in H_p(P))$ in $H_p(M)$ under the map induced by the inclusion, and it is often still denoted by $[P]$. When $P$ is a compact oriented $p$-dimensional manifold with boundary embedded in a topological space $X$ so that $\partial P$ is embedded in a subspace $Y$ of $X$, we define the class $[P, \partial P]$ of $(P, \partial P)$ in $H_p(X, Y)$, similarly.

With these conventions, when $M$ is a connected, compact, oriented, $n$-dimensional manifold with boundary, such that $M$ can be equipped with a triangulation, the boundary map $\partial_n$ in the homology sequence of the pair $(M, \partial M)$ maps $[M, \partial M]$ to the class $[\partial M]$, where $[\partial M]$ is the sum of the classes of the connected components of $\partial M$ equipped with the orientation induced by the orientation of $M$ with respect to the outward normal first convention.

All the manifolds considered in this book can be equipped with triangulations. If $M$ is a manifold equipped with a triangulation $T$, and if $P$ is a $p$-dimensional closed oriented manifold embedded in $M$ that is a union of simplices of $T$, then the homology class of $P$ in $M$ vanishes if and only if the cycle that is the sum of the simplices of dimension $p$ of $P$ (equipped with the orientation of $P$) is the (algebraic) boundary of a simplicial chain of $T$ of dimension $p+1$.

Homologies with various coefficients are related by the universal coefficient theorem. See [Gre67, 29.12], [Hat02, Thm. 3A.3], or [Spa81, Chap. 5, Sec.2], for example. Here is an excerpt of this theorem.

**Theorem A.7.** When $\Lambda = \mathbb{Q}$ or $\mathbb{R}$, we have

$$H_k(X; \Lambda) = H_k(X; \mathbb{Z}) \otimes_{\mathbb{Z}} \Lambda$$

for any topological space $X$.

In this book, we mostly use cohomology with coefficients in $\mathbb{Q}$, $\mathbb{Z}/2\mathbb{Z}$ or $\mathbb{R}$. In these cases, it can be defined by the following excerpt of the universal coefficient theorem for cohomology, which can be found in [Spa81, Chap.5, Sec.5, Thm. 3, page 243] and in [Hat02, Thm. 3.2], for example.
**Theorem A.8.** When $\Lambda$ is a field, we have

$$H^k(X, Y; \Lambda) = \operatorname{Hom}_{\Lambda}(H_k(X, Y; \Lambda), \Lambda)$$

for any $k \in \mathbb{Z}$ and for any topological space $X$.

Note the sign = in the above theorems, meaning that the identifications are canonical. For a continuous map $f: (X, Y) \to (A, B)$ and an integer $k \in \mathbb{Z}$, the morphism

$$H^k(f; \Lambda): H^k(A, B; \Lambda) \to H^k(X, Y; \Lambda)$$

maps a linear map $g$ of $H^k(A, B; \Lambda)$ to $g \circ H_k(f; \Lambda)$.

For a general $\Lambda$, and for a pair $(X, Y)$ of topological spaces such that $X$ is equipped with a triangulation $T$ as above, and $Y$ is equipped with a subtriangulation $T_Y$ of $T$ as before, the cohomology $H^*(X, Y; \Lambda)$ of $(X, Y)$ is the cohomology of the complex $(C^*(T, T_Y; \Lambda), \partial^*(T, T_Y; \Lambda))$, where $C^k(T, T_Y; \Lambda)$ is equal to $\operatorname{Hom}(C_k(T, T_Y; \Lambda); \Lambda)$ and $\partial^k: C^k(T, T_Y; \Lambda) \to C^{k+1}(T, T_Y; \Lambda)$ maps a linear form $g$ to $g \circ \partial_{k+1}$. We have

$$H^k(X, Y; \Lambda) = \frac{\operatorname{Ker}(\partial^k)}{\operatorname{Im}(\partial^{k-1})}.$$

The elements of $\operatorname{Ker}(\partial^k)$ are the $k$-dimensional cocycles, and the elements of $\operatorname{Im}(\partial^{k-1})$ are the $k$-dimensional coboundaries.

Here is a weak version of the Poincaré duality theorem. See [Gre67, Chap. 26 to 28, in particular (28.18)] or [Hat02, Thm. 3.43].

**Theorem A.9.** Let $M$ be a compact, $n$-dimensional manifold with a possible boundary. Let $H$ denote the singular homology. Then there are canonical Poincaré duality isomorphisms from $H^k(M, \partial M; \mathbb{Z}/2\mathbb{Z})$ to $H_{n-k}(M; \mathbb{Z}/2\mathbb{Z})$, and from $H^k(M; \mathbb{Z}/2\mathbb{Z})$ to $H_{n-k}(M, \partial M; \mathbb{Z}/2\mathbb{Z})$, for any $k \in \mathbb{Z}$.

If $M$ is oriented, then for any $\Lambda \in \mathbb{Z}/2\mathbb{Z}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$, there are canonical Poincaré duality isomorphisms from $H^k(M, \partial M; \Lambda)$ to $H_{n-k}(M; \Lambda)$, and from $H^k(M; \Lambda)$ to $H_{n-k}(M, \partial M; \Lambda)$.

Let $A$ be an oriented smooth submanifold of $M$ of dimension $n-k$ such that $\partial A \subset \partial M$. Let $P^{-1}([A, \partial A]) \in H^k(M; \Lambda)$ denote the image of its class under the inverse of such a Poincaré duality isomorphism. Let $B$ be the class of a closed oriented $k$-dimensional submanifold or a simplicial $k$-dimensional cycle transverse to $A$. Then the evaluation $P^{-1}([A, \partial A])(B)$ of $P^{-1}([A, \partial A])$ at $B$ is the algebraic intersection $\langle B, A \rangle_M$.

Here is a weak version of the Künneth theorem. See [Spa81, Chap.5, Sec.3, Thm. 10 (and Chap.5, Sec.2, Lem. 5)] or [Hat02, Section 3.B].
**Theorem A.10.** Let $H$ denote the singular homology with coefficients in a commutative field $\Lambda$. Then for any two topological spaces $X$ and $Y$, for any $k \in \mathbb{N}$, we have

$$H_k(X \times Y) = \bigoplus_{i=0}^{k} H_i(X) \otimes_{\Lambda} H_{k-i}(Y).$$

Again, the sign = means that the identification is canonical. For embeddings of oriented closed manifolds $P$ into $X$ and $Q$ into $Y$, the tensor product $[P] \otimes [Q]$ of the homology classes of their images, represents the homology class of $P \times Q$.

We end this section by stating a weak version of the following Mayer–Vietoris exact sequence, which can be recovered from the Eilenberg–Steenrod axioms. See [Gre67, Chap. 17, Thm. 17.6], for example.

**Theorem A.11.** Let $X$ be a topological space. Let $A$ and $B$ be subspaces of $X$ whose union $A \cup B$ equals $X$. Let $i_A: A \cap B \hookrightarrow A$, $i_B: A \cap B \hookrightarrow B$, $j_A: A \hookrightarrow X$, $j_B: B \hookrightarrow X$ denote the inclusion maps. Assume that $j_A$ and $j_B$ induce isomorphisms from $H(A, A \cap B)$ to $H(X, B)$ and from $H(B, A \cap B)$ to $H(X, A)$. Let $\partial_{MV,k+1}: H_{k+1}(X) \to H_k(A \cap B)$ be the composition of the map from $H_{k+1}(X)$ to $H_{k+1}(X, B)$ induced by inclusion, the inverse of the isomorphism from $H_{k+1}(A, A \cap B)$ to $H_{k+1}(X, B)$, and the boundary map $\partial_{k+1}(A, A \cap B): H_{k+1}(A, A \cap B) \to H_k(A \cap B)$ of the long exact sequence of $(A, A \cap B)$. Then there is a long exact sequence

$$\dots \to H_{k+1}(X) \xrightarrow{\partial_{MV,k+1}} H_k(A \cap B) \xrightarrow{i_{MV,k}} H_k(A) \oplus H_k(B) \xrightarrow{j_{MV,k}} H_k(X) \xrightarrow{\partial_{MV,k}} \dots$$

such that $j_{MV,k}(\alpha \in H_k(A), \beta \in H_k(B)) = H_k(j_A)(\alpha) + H_k(j_B)(\beta)$ and $i_{MV,k}(\gamma) = (H_k(i_A)(\gamma), -H_k(i_B)(\gamma))$.

## A.2 Homotopy groups

Let $X$ be a topological space equipped with a basepoint $x$. Let $n$ be a positive integer. The set of homotopy classes of maps from $[0, 1]^n$ to $X$ mapping $\partial([0, 1]^n)$ to $x$ is denoted by $\pi_n(X, x)$. It is a homotopy group. Its product sends a pair $([f], [g])$ of homotopy classes of maps $f$ and $g$ such that $f$ sends $[1/2, 1] \times [0, 1]^{n-1}$ to $x$ and $g$ sends $[0, 1/2] \times [0, 1]^{n-1}$ to $x$ to the class $[f] [g]$ of the map that coincides with $f$ on $[0, 1/2] \times [0, 1]^{n-1}$ and with $g$ on $[1/2, 1] \times [0, 1]^{n-1}$. This product is commutative when $n \ge 2$. The set of path-connected components of $X$ is denoted by $\pi_0(X)$.

**Remark A.12.** Classically, the set $\pi_n(X, x)$ is defined as the set of homotopy classes of maps from $(S^n, (1, 0, \dots, 0))$ to $(X, x)$. The two definitions coincide since $S^n$ is homeomorphic to the quotient $[0, 1]^n/(\partial[0, 1]^n)$ for $n \ge 1$.
**Examples A.13.** Let $k$ and $n$ be positive integers, such that $1 \le k \le n$. A standard approximation theorem [Hir94, Chapter 2, Theorem 2.6, p. 49] implies that any continuous map from $S^k$ to $S^n$ is homotopic to a smooth map. The Morse–Sard theorem 1.4 ensures that if $k < n$, any smooth map is valued in the complement of a point in $S^n$, which is contractible. Therefore, $\pi_k(S^n, *) = \{1\}$ if $1 \le k < n$. The reader can develop these arguments to prove that there is a canonical isomorphism from $\pi_n(S^n, *)$ to $\mathbb{Z}$, which maps the homotopy class of a smooth map from $(S^n, *)$ to itself to its degree (see Definition 1.5).

A weak version of the *Hurewicz theorem* relating homotopy groups to homology groups, ensures that for any path-connected topological space $X$ equipped with a basepoint $x$, $H_1(X; \mathbb{Z})$ is the abelianization of $\pi_1(X, x)$. See [Gre67, (12.1)], for example.

A map $p: E \to B$ is called a *weak fibration* if it satisfies the following homotopy lifting property with respect to cubes:

For any integer $n \in \mathbb{N}$, for any pair $(h_0: [0, 1]^n \times \{0\} \to E, H: [0, 1]^{n+1} \to B)$ of continuous maps such that $H|_{[0,1]^n \times \{0\}} = p \circ h_0$, there exists a continuous extension $h$ of $h_0$ to $[0, 1]^{n+1}$ such that $H = p \circ h$.

To such a weak fibration, we associate the following long exact sequence in homotopy [Spa81, Chap. 7, Sec. 2, Thm. 10].

**Theorem A.14.** Let $p: E \to B$ be a weak fibration. Let $e \in E$ be a basepoint of $E$. Let $b = p(e)$ denote its image under $p$, and let $F = p^{-1}(b)$ denote the fiber over $b$. Then we have the long exact sequence

$$ \dots \pi_{n+1}(B, b) \to \pi_n(F, e) \to \pi_n(E, e) \to \pi_n(B, b) \to \pi_{n-1}(F, e) \dots \\ \dots \to \pi_1(B, b) \to \pi_0(F) \to \pi_0(E) \to \pi_0(B), $$

where the maps between the $\pi_n$ are respectively induced by the inclusion $F \hookrightarrow E$ and by $p$, and the map from $\pi_n(B, b) \to \pi_{n-1}(F, e)$ is constructed as follows. An element of $\pi_n(B, b)$ is represented by a map $H: [0, 1]^n \to B$, which has a lift $h: [0, 1]^n \to E$ that maps $[0, 1]^{n-1} \times \{0\} \cup (\partial[0, 1]^{n-1} \times [0, 1])$ to $e$. Such an element is mapped to the homotopy class of the restriction of $h$ to $[0, 1]^{n-1} \times \{1\}$. The last three maps of the exact sequence are just maps between sets. Exactness means that the preimage of the component of the basepoint is the image of the previous map.

**Remark A.15.** To define the map from $\pi_n(B, b)$ to $\pi_{n-1}(F, e)$ in the above theorem, we implicitly used the fact that the pair

$$ ([0, 1]^n, [0, 1]^{n-1} \times \{0\} \cup (\partial[0, 1]^{n-1} \times [0, 1])) $$

is homeomorphic to $([0, 1]^n, [0, 1]^{n-1} \times \{0\})$ for all $n \in \mathbb{N} \setminus \{0\}$.
A path from a point $x$ to another point $y$ of $X$ induces an isomorphism
from $\pi_n(X, x)$ to $\pi_n(X, y)$ for any integer $n$. So the basepoint is frequently
omitted from the notation $\pi_n(X, x)$ when $X$ is path-connected.

A map $p: E \to B$ is called a *covering map* if every $b \in B$ has a neighborhood $U$ such that $p^{-1}(U)$ is a disjoint union of subsets of $E$, each of which is mapped homeomorphically onto $U$ by $p$. Such a covering map is an example of a weak fibration, for which $p$ induces isomorphisms from $\pi_n(E, e)$ to $\pi_n(B, p(e))$ for any $e \in E$ and any $n \ge 2$.
# Appendix B

## Differential forms and de Rham cohomology

Here are a few well-known results about differential forms used throughout the book. More detailed accounts can be found in the books [God71, Chapters XI and XII] by Claude Godbillon and [BT82] by Raoul Bott and Loring Tu.

### B.1 Differential forms

Let $M$ be a smooth manifold with possible boundary and ridges. Degree 0 forms of $M$ are smooth functions on $M$. The differential $df = Tf: TM \to \mathbb{R}$ of a smooth map $f$ from $M$ to $\mathbb{R}$ is an example of a degree-one form of $M$. In general, a degree $k$ differential form on $M$ is a smooth section of the bundle $\wedge^k(TM)^* = \text{Hom}(\wedge^k TM; \mathbb{R})$ over $M$. So, such a form $\omega$ maps any $m \in M$ to an element $\omega(m)$ of $\text{Hom}(\wedge^k T_m M; \mathbb{R})$, smoothly, in the sense below. On an open part $U$ of a manifold identified with an open subspace of $\mathbb{R}^n$ by a chart $\phi: U \to \mathbb{R}^n$ mapping $u \in U$ to $\phi(u) = (\phi_1(u), \dots, \phi_n(u))$, the degree $k$ forms are uniquely expressed as

$$ \sum_{(i_1, \dots, i_k) \in \mathbb{N}^k: 1 \le i_1 < \dots < i_k \le n} f_{i_1 \dots i_k} d\phi_{i_1} \wedge \dots \wedge d\phi_{i_k}. $$

for smooth maps $f_{i_1 \dots i_k}: U \to \mathbb{R}$. The vector space of degree $k$ differential forms on $M$ is denoted by $\Omega^k(M)$. The differential $T\psi: TM \to TN$ of a smooth map $\psi$ from a manifold $M$ to a manifold $N$ induces the *pull-back* of differential forms

$$ \psi^*: \Omega^k(N) \to \Omega^k(M) \\ \omega \mapsto (x \mapsto \omega(\psi(x)) \circ \wedge^k T_x \psi). $$
The antisymmetric bilinear exterior product $\wedge$ equips $\oplus_{k \in \mathbb{N}} \Omega^k(M)$ with a structure of graded algebra, such that $\psi^*(\omega \wedge \omega') = \psi^*(\omega) \wedge \psi^*(\omega')$ for any two forms $\omega$ and $\omega'$ on $N$. This graded algebra is equipped with a unique operator $d: \Omega^k(M) \to \Omega^{k+1}(M)$ such that

* $(df = Tf)$ is the differential of $f$ for any $f \in \Omega^0(M)$,

* we have $d \circ d(f) = 0$ for any $f \in \Omega^0(M)$,

* for any $\alpha \in \Omega^{|\alpha|}(M)$ and $\beta \in \Omega^{|\beta|}(M)$, we have

$$d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^{|\alpha|} \alpha \wedge d\beta,$$

where $|\alpha|$ and $|\beta|$ denote the respective integral degrees of $\alpha$ and $\beta$, and

* the derivation operator $d$ commutes with the above pull-backs, we have $d\psi^*(\omega) = \psi^*(d\omega)$.

Then we have $d \circ d = 0$.

The support of a differentiable form is the closure of the set where it does not vanish.

Let $x_i: \mathbb{R}^k \to \mathbb{R}$ be the usual coordinate functions. The integral of a degree $k$ differential form $\omega = fdx_1 \wedge \cdots \wedge dx_k$ over a $k$-dimensional compact submanifold $C$ of $\mathbb{R}^k$ with boundary and ridges (like $[0, 1]^k$) is $\int_C \omega = \int_C fdx_1 \dots dx_k$. For any smooth map $\psi$ of $\mathbb{R}^k$ that restricts to an orientation-preserving diffeomorphism from $C$ to its image, we have

$$\int_C \psi^*(\omega) = \int_{\psi(C)} \omega.$$

Thanks to this property, we unambiguously define the integral of a $k$-form $\omega$ over any $k$-dimensional compact submanifold $C$ of a manifold $M$, identified to a subspace of $\mathbb{R}^k$ by a diffeomorphism $\phi: C \to \mathbb{R}^k$ onto its image, to be

$$\int_C \omega = \int_{\phi(C)} \phi^{-1*}(\omega).$$

This definition extends naturally to general compact manifolds with boundaries and ridges.

One of the most useful theorems in this book is the following Stokes theorem. See [BT82, Theorem 3.5, Page 31].
**Theorem B.1** (Stokes' theorem). Let $\omega$ be a degree $d$ form on an oriented smooth compact $(d+1)$-manifold $M$ with boundary $\partial M$, then we have

$$ \int_{\partial M} \omega = \int_{M} d\omega. $$

This theorem applies to manifolds with ridges, and $\int_{\partial M} \omega$ is the sum of the $\int_{C} \omega$, over the codimension zero faces $C$ of $\partial M$. (The closures of these faces are $d$-manifolds with boundaries.)

## B.2 De Rham cohomology

A degree $k$ differential form $\omega$ on $M$ is closed if $d\omega = 0$. It is exact if $\omega \in d\Omega^{k-1}(M)$. Define the degree $k$ de Rham cohomology module of $M$ to be

$$ H_{dR}^{k}(M) = \frac{\text{Ker} (d: \Omega^{k}(M) \to \Omega^{k+1}(M))}{d(\Omega^{k-1}(M))}. $$

For a compact submanifold $N$ (without boundary) of the interior of $M$, or for a connected component $N$ of $\partial M$, as in [God71, Chapter XII], let $\Omega^k(M, N)$ be the kernel of the restriction map from $\Omega^k(M)$ to $\Omega^k(N)$. The relative degree $k$ de Rham cohomology module of $(M, N)$ is

$$ H_{dR}^{k}(M, N) = \frac{\text{Ker} (d: \Omega^{k}(M, N) \to \Omega^{k+1}(M, N))}{d(\Omega^{k-1}(M, N))}. $$

We have a natural short exact sequence of chain complexes

$$ 0 \to \Omega^k(M, N) \to \Omega^k(M) \to \Omega^k(N) \to 0. $$

This sequence induces a natural long exact cohomology sequence

$$ \to H_{dR}^{k-1}(N) \to H_{dR}^{k}(M, N) \to H_{dR}^{k}(M) \to H_{dR}^{k}(N) \to $$

where the maps from $H_{dR}^k(M, N)$ to $H_{dR}^k(M)$ and from $H_{dR}^k(M)$ to $H_{dR}^k(N)$ are induced by the restrictions and the cohomology class $[\eta]$ in $H_{dR}^{k-1}(N)$ of a closed form $\eta$ of $N$ is mapped to $([d\tilde{\eta}] \in H_{dR}^k(M, N))$ for an extension $\tilde{\eta}$ of $\eta$ to $M$.

The degree $k$ forms with compact support in $M$ also form a subspace $\Omega_c^k(M)$ of $\Omega^k(M)$, and the degree $k$ de Rham cohomology module with compact support of $M$ is

$$ H_{dR,c}^{k}(M) = \frac{\text{Ker} (d: \Omega_{c}^{k}(M) \to \Omega_{c}^{k+1}(M))}{d(\Omega_{c}^{k-1}(M))}. $$
For any smooth map $\psi$ from $M$ to another manifold $M'$ that maps $N$ to a submanifold $N'$, the pull-back $\psi^*$ induces maps still denoted by $\psi^*$ from $H_{dR}^k(N')$ to $H_{dR}^k(N)$, and from $H_{dR}^k(M', N')$ to $H_{dR}^k(M, N)$. If $\phi$ is another such smooth map from the pair $(M', N')$ to another such $(M'', N'')$, then we have

$$ \psi^* \circ \phi^* = (\phi \circ \psi)^*. $$

When such a map $\psi$ is proper (i.e., when the preimage of a compact is compact), the map $\psi$ also induces $\psi^*$: $H_{dR,c}^k(M') \to H_{dR,c}^k(M)$.

The following standard lemma implies that $\psi^*$: $H_{dR}^k(M') \to H_{dR}^k(M)$ depends only on the homotopy class of $\psi$: $M \to M'$.

**Lemma B.2.** Let $V$ and $W$ be two smooth manifolds, and let

$$ h: \begin{array}{ccc} [0, 1] \times V & \to & W \\ (t, v) & \mapsto & h_t(v) \end{array} $$

be a smooth homotopy. Let $\omega$ be a degree $d$ closed form on $W$. Then we have

$$ h_t^*(\omega) - h_0^*(\omega) = d\eta_t(h, \omega) $$

for any $t \in [0, 1]$, where $\eta_t(h, \omega)$ is the following degree ($d-1$) form on $V$. For $u \in [0, 1]$, let $i_u: V \to [0, 1] \times V$ map $v \in V$ to $i_u(v) = (u, v)$. Let $h^*(\omega)((u, v))(\frac{\partial}{\partial t} \wedge \cdot)$ be obtained from $h^*(\omega)$ by evaluating it at the tangent vector $\frac{\partial}{\partial t}$ to $[0, 1] \times \{v\}$ at $(u, v)$. (Thus, $h^*(\omega)((u, v))(\frac{\partial}{\partial t} \wedge \cdot)$ is the value at $(u, v) \in [0, 1] \times V$ of a degree ($d-1$) form of $[0, 1] \times V$.) Then we define

$$ \eta_t(h, \omega)(v) = \int_0^t i_u^* \left( h^*(\omega)((u, v)) \left( \frac{\partial}{\partial t} \wedge \cdot \right) \right) du. $$

**PROOF:** Observe $h_u^*(\omega) = i_u^*(h^*(\omega))$ and write

$$ h^*(\omega) = \omega_1 + dt \wedge h^*(\omega) \left( \frac{\partial}{\partial t} \wedge \cdot \right), $$

where we have $h_u^*(\omega) = i_u^*(\omega_1)$. Since $\omega$ is closed, $dh^*(\omega)$ vanishes. Therefore, we obtain

$$ 0 = d\omega_1 - dt \wedge d\left(h^*(\omega)\left(\frac{\partial}{\partial t} \wedge \cdot\right)\right). $$

On the other hand, with the natural projection $p_V: [0, 1] \times V \to V$, we can write

$$ (d\omega_1)((u, v)) = \left( dt \wedge \frac{\partial}{\partial t}(\omega_1) + p_V^* (dh_u^*(\omega)) \right) ((u, v)). $$
We get

$$i_u^* \left( \frac{\partial}{\partial t}(\omega_1) \right) = i_u^* \left( d \left( h^*(\omega) \left( \frac{\partial}{\partial t} \wedge \cdot \right) \right) \right)$$

and

$$\frac{\partial}{\partial u} h_u^*(\omega) = \frac{\partial}{\partial u} i_u^*(\omega_1) = i_u^* \left( \frac{\partial}{\partial t}(\omega_1) \right) = i_u^* \left( d \left( h^*(\omega) \left( \frac{\partial}{\partial t} \wedge \cdot \right) \right) \right).$$

Since we have $h_t^*(\omega)(v) - h_0^*(\omega)(v) = \int_0^t \frac{\partial}{\partial u} h_u^*(\omega)(v) du$, the lemma follows. □

In particular, if there exist smooth maps $f: M \to N$ and $g: N \to M$ such that $f \circ g$ is smoothly homotopic to the identity map of $N$, and $g \circ f$ is smoothly homotopic to the identity map of $M$, then $f^*$ is an isomorphism from $H_{dR}^k(N)$ to $H_{dR}^k(M)$ for any $k$. In particular, all the smoothly contractible manifolds, such as $\mathbb{R}^n$, have the same de Rham cohomology as the point. So, for such a manifold $C$, $H_{dR}^k(C) = \{0\}$ for any $k \neq 0$, and $H_{dR}^0(C) = \mathbb{R}$.

More generally, in 1931, Georges de Rham identified the de Rham cohomology of a smooth $n$-dimensional manifold $M$ to its singular cohomology with coefficients in $\mathbb{R}$. See [War83, Pages 205-207], or [BT82, Theorems 8.9 Page 98, 15.8 page 191].

The de Rham isomorphism sends the cohomology class $[\omega]$ of a closed degree $k$ form to a linear map $[\omega]_{dR}$ of $\text{Hom}(H_k(M; \mathbb{R}); \mathbb{R}) = H^k(M; \mathbb{R})$. The linear map $[\omega]_{dR}$ sends the homology class of a closed oriented $k$-dimensional submanifold $N$ of $M$ to $[\omega]_{dR}([N]) = \int_N \omega$.

If $M$ has a smooth triangulation $T$, any $k$-form $\omega$ similarly defines a simplicial cochain, i.e., an element of $C^k(T; \mathbb{R}) = \text{Hom}(C_k(T; \mathbb{R}); \mathbb{R})$, by integration over the simplices. Stokes' theorem guarantees that the induced map from $\Omega^k(M)$ to $C^k(T; \mathbb{R})$ commutes with differentials, and induces a morphism from $H_{dR}^k(M)$ to $H^k(M; \mathbb{R}) = \text{Hom}(H_k(M; \mathbb{R}); \mathbb{R})$.

We also have the following theorem [GHV72, page 197] or [BT82, (5.4) and Remark 5.7].

**Theorem B.3.** For any oriented manifold $M$ without boundary of dimension $n$ (whose cohomology is not necessarily finite-dimensional), the morphism from $\Omega^k(M; \mathbb{R})$ to $\text{Hom}(\Omega_c^{n-k}(M); \mathbb{R})$

$$(\omega \mapsto (\omega_2 \mapsto \int_M \omega \wedge \omega_2))$$

induces an isomorphism from $H^k(M; \mathbb{R})$ to $\text{Hom}(H_c^{n-k}(M); \mathbb{R})$, for any integer $k$.
In particular, when the real cohomology of $M$ is finite-dimensional, so is its homology, and $H_c^{n-k}(M)$ is isomorphic to $H_k(M; \mathbb{R})$. Below, we exhibit the image $[\omega_A]$ of the homology class of an oriented compact submanifold $A$ with trivial normal bundle in $H_k(M; \mathbb{R}) = \operatorname{Hom}(H^k(M); \mathbb{R})$ under a canonical isomorphism from $H_k(M; \mathbb{R})$ to $H_c^{n-k}(M)$.

Let $A$ be such a $k$-dimensional submanifold without boundary of the manifold $M$, and let $N(A) = B^{n-k} \times A$ be a tubular neighborhood of $A$ in $M$. Let $\omega_B$ be an $(n-k)$-form of $B^{n-k}$ supported in the interior of $B^{n-k}$ such that $\int_{B^{n-k}} \omega_B = 1$. Use the natural projection $p_B: B^{n-k} \times A \to B^{n-k}$ to pull back $\omega_B$ on $N(A)$ and define a form $\omega_A$ to coincide with $p_B^*(\omega_B)$ on $N(A)$ and to be zero outside $N(A)$. The cohomology class $[\omega_A]$ of $\omega_A$ is independent of the choice of $\omega_B$. The integral of the closed form $\omega_A$ over a compact $(n-k)$ submanifold $B$ of $M$ transverse to $A$ is the algebraic intersection $\langle B, A \rangle_M$ of $B$ and $A$. Note that the support of $\omega_A$ may be chosen in an arbitrarily small neighborhood of $A$. In the words of Definition 11.6, the form $\omega_A$ may be chosen to be $\alpha$-dual to $A$ for an arbitrarily small positive number $\alpha$.

**Lemma B.4.** Let $M$ be a compact manifold of dimension $n$ with possible boundary. Assume that $M$ is equipped with a smooth triangulation. The above process can be extended to produce canonical Poincaré duality isomorphisms from $H_a(M, \partial M; \mathbb{R})$ to $H_{\partial M}^{n-a}(M)$, where such a Poincaré duality isomorphism maps the class of an $\alpha$-dimensional cycle of $(M, \partial M)$ to the class of a closed $(n-a)$-form $\omega_A$ $\alpha$-dual to $A$ for an arbitrarily small positive number $\alpha$, as follows.

**PROOF:** Let $A$ be a linear combination of $\alpha$-dimensional simplices of the triangulation $T$ of $M$ such that the algebraic boundary of $A$ is in $C(T_{\partial M})$. Such an $A$ is a simplicial $\alpha$-cycle of $(M, \partial M)$. Its support $\bar{A}$ is the union in $M$ of the closed simplices with a nonzero coefficient of $A$. We construct a form $\omega_A$ $\alpha$-dual to $A$ as follows. Let $A^{(k)}$ be the intersection of $\bar{A}$ with the $k$-skeleton of $T$. ($A^{(k)} = \emptyset$ if $k < 0$.) Let $N(A^{(a-1)})$ be a small neighborhood of $A^{(a-1)}$.

First construct a closed form $\omega_A$ $\alpha$-dual to $A$ outside $N(A^{(a-1)})$ so that its support is in an arbitrarily small neighborhood of $A$, as explained above in the case of closed manifolds with trivial normal bundles.

Then extend $\omega_A$ around each $(a-1)$-simplex $\Delta$ of $A$, outside a small neighborhood $N(A^{(a-2)})$ of $A^{(a-2)}$ as follows.

Without loss of generality, assume that the intersection of a neighborhood of $\Delta$ with the complement of $N(A^{(a-2)})$ is diffeomorphic to $\Delta' \times D^{n-a+1}$, for some interior $\Delta'$ of an $(a-1)$-simplex in the interior of $\Delta$ (some truncation of $\Delta$). See Figure B.1. The $(n-a)$-form $\omega_A$ is defined and closed on a neighborhood of $\Delta' \times \partial D^{n-a+1}$. Its integral along $\{x\} \times \partial D^{n-a+1}$ is the
algebraic intersection $\langle \partial D^{n-a+1}, A \rangle_M$. It is the coefficient of $\Delta$ in $\partial A$, up to sign. Since this coefficient is zero, the form $\omega_A$ is exact in a neighborhood of $\Delta' \times \partial D^{n-a+1}$. So $\omega_A$ may be written as $d\eta$ on this neighborhood. Choose a map $\chi$ on $\mathbb{R}^{n-a+1}$ that takes the value 1 outside a small neighborhood of 0 and that vanishes in a smaller neighborhood of 0 so that $d\chi\eta$ makes sense on $\Delta' \times D^{n-a+1}$ and extends $\omega_A$ as a closed form.

<figure><img src="image_168.png" /><figcaption>Figure B.1: The neighborhoods $N(A^{(1)})$ and $N(A^{(0)})$, a simplex $\Delta$ and its truncation $\Delta'$, when $A$ is one-dimensional</figcaption></figure>

This process allows us to define a closed form $\omega_A$ $\alpha$-dual to $A$ outside a small neighborhood $N(A^{(a-2)})$ of $A^{(a-2)}$. Iterate the process in order to extend such a form outside $N(A^{(a-3)})$, outside $N(A^{(a-4)})$, ..., and on the whole $M$. Note that the forms will be automatically exact around the truncated smaller cells since we have $H^{n-a}(\partial D^{n-a+k}) = 0$ when $k > 1$. $\square$

In this setting, the correspondence between chain boundaries and the differentiation operator $d$ can be roughly seen as follows. Let $A$ be a compact oriented $\alpha$-dimensional submanifold with boundary of a manifold $M$. Assume that the normal bundle of $A$ is trivial. Let $[-1, 1] \times \partial A$ be embedded in the interior of $M$ so that $\{0\} \times \partial A = \partial A$ and $[-1, 0] \times \partial A$ is a neighborhood of $\partial A$ in $A$. Set $A^+ = A \cup_{[-1,0] \times \partial A} [-1, 1] \times \partial A$. Choose a map $\chi: A^+ \to [0, 1]$ sending $A$ to 1 and a neighborhood of $\partial A^+$ to 0 such that $\chi$ factors through the projection onto $[0, 1]$ on $[0, 1] \times \partial A$. Let $N(A^+) = B^{n-a} \times A^+$ be the total space of the normal bundle to $A^+ = A \cup_{\partial A} ([0, 1] \times \partial A)$ embedded in $M$. Let $p_{A^+}: N(A^+) \to A^+$ and $p_B: N(A^+) \to B^{n-a}$ denote the natural projections. Let $\omega_B$ be an $(n-a)$-form of $B^{n-a}$ supported in the interior of $B^{n-a}$ such that $\int_{B^{n-a}} \omega_B = 1$. The form

$$ \omega_A = (\chi \circ p_{A^+}) p_B^*(\omega_B) $$

extends as a smooth form that vanishes outside $N(A^+)$. The form $d\omega_A$ is supported in $B^{n-a} \times [0, 1] \times \partial A$. Its integral along a chain $C$ whose boundary does not meet $B^{n-a} \times [0, 1] \times \partial A$ is $\pm \langle C, \partial A \rangle_M$.
<figure><img src="image_169.png" /></figure>
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<figure><img src="image_170.png" /></figure>
Terminology

A-holonomy, 378, 382

algebraic boundary, 568

algebraic intersection, 34, 58

algebraically trivial
one-cycle, 162

anomaly, 224, 226

antisymmetric
propagating form, 75

A-numbered Jacobi diagram, 148,
377

associator, 283

asymptotic rational homology $\mathbb{R}^3$,
73

asymptotically standard
parallelization, 72

automorphism
of a Jacobi diagram, 158

based $\Delta$-parenthesization, 186

blow-up, 66

blowdown map, 66

boundary map, 568

braid, 288

braid representative, 287

cabling, 289

chain, 58

chord diagram, 121, 448

closed
form, 577
manifold, 39

coboundary, 571

cocycle, 571

codimension *j* boundary, 56

codimension-one system of edges,
311

configuration, 22, 26, 145
space, 22, 26, 42

connected sum, 55

consecutive elements, 179

coorientation, 57

coproduct, 139

count, 139

covering map, 574

cycle, 58, 569

daughter, 172, 185

de Rham cohomology, 577

degree
at a point, 20
of a Jacobi diagram, 128
of a map, 20
of an invariant, 119

Dehn surgery, 460

dilation, 152

edge-orientation, 146

exact
form, 577
sequence, 564
exterior of a knot, 460

finite type invariant, 45

framed
link, 47

tangle, 275, 279

framing

of a knot, 164

general $3n$ position, 245

group-like, 140

holonomy, 368

homogeneous

propagating form, 74, 509

homotopy, 62

homotopy equivalence, 62

homotopy group, 572

horizontally normalizing, 328, 349

integer homology sphere, 16

Jacobi diagram, 42, 127

kid, 172, 186

Lagrangian, 46

Lagrangian-Preserving surgery, 46

leg, 491

linking number, 34, 60

long tangle representative, 269

longitude, 461

LP surgery, 46

LTR, 269

maximal free system of edges, 309

meridian of a knot, 61

mother, 186

orientation

of a manifold, 18

of a set, 146

of a trivalent vertex, 127

of a univalent vertex, 127

of a vector space, 18

oriented

Jacobi diagram, 127

parallel

of a knot, 25

of a long component, 275

parallelization

of a 3-manifold, 63

parenthesization, 172

$\Delta$-parenthesization, 185

Pontrjagin class

first, 97

Pontrjagin number, 98

possibly separating set, 324

preferred longitude of a knot, 461

primitive, 140

propagating

chain, 73, 74, 560

form, 73, 74, 553

propagator, 73

q-braid, 288

Q-cylinder, 15

Q-sphere, 15, 16

$q$-tangle, 288

quaternions, 100

rational homology

cylinder, 267

sphere, 16

reducing system of edges, 309

regular point, 19

regular value

for a general domain, 20

of a smooth map, 19

relative

Euler number, 510

ridges, 18

Seifert surface, 464

self-linking number, 275

semi-algebraic set, 394

small diagonal, 174

son, 172

special
complex trivialization, 96

straight

link, 164

tangle, 386

support

of a form, 576

symplectic basis, 464

tangle, 278

representative, 277

$\tau$-bounding, 503

$T$-faces, 322

transverse

intersection, 34

map, 244

unit normal bundle, 65

universal

finite type invariant, 474

Vassiliev knot invariant, 125

Vassiliev invariant, 45

vertex-orientation

of $H(\Gamma)$, 147

of a Jacobi diagram, 127

vertically normalizing, 328, 349

volume-one form of $S^2$, 73

weight system, 128

$\mathbb{Z}$-sphere, 15
<figure><img src="image_171.png" /></figure>
Index of notation

$A_I^{(i)}$ 3-handlebody, 483

$A_t^{(i)}$ 3-handlebody, 489

$\mathcal{A}_n(.)$ target quotiented diagram
space of $\mathcal{Z}_n$, 128, 129

$\tilde{\mathcal{A}}_n(.)$ no trivalent component, 130

$\mathcal{A}_n^c(\emptyset)$ connected, 154

$\mathcal{A}_n$ chord diagrams mod 4T, 124

$\alpha$ anomaly, 157

$b(A)$ basepoint, 186

$B(A, C)$ blow-up, 66

$\tilde{B}_{1,\infty}$ ball around $\infty$, 70

$B_R$ rational homology ball, 73

$\beta$ anomaly, 223

Configuration spaces

$C_2(R)$, 70

$\tilde{C}_V[M]$ open, 191

$\tilde{C}(R, L; \Gamma)$ open, 145

Daughter sets

$D(V)$, 172

$D(V, K(V))$, 172

$D_\epsilon$ disk of radius $\epsilon$, 258

$D(\omega_0(R_I))$ domain in $C_2$, 533

Diagram sets

$\mathcal{D}_n^c$ connected, 154

$\mathcal{D}_{n,A}^e(\mathcal{L})$ A-numbered, 160, 377

$\mathcal{D}_{n,m}^e(\mathcal{L})$, 160

$\mathcal{D}_n^e \underline{3n}$-numbered, 148

$\mathcal{D}_{3n}^e \underline{3n}$-numbered, 245

$\mathcal{D}_n^u$ unnumbered, 158

Diagonals

$\Delta((\mathbb{R}^3)^2)$, 62

$\Delta_A(X^V)$, 174

$\partial_j$ codimension $j$ boundary, 56

$E(\Gamma) = \{\text{edges of } \Gamma\}$, 42

$\mathbf{e}(\mathcal{E}, l)$ edge in a tree, 308

$\eta_{[-1,1]}$, 483

$\eta(a_j^i)$ one-form on $A_I^{(i)}$, 483

Faces

$F(A, L, \Gamma)$, 179

$F_\infty(A, L, \Gamma)$, 182

$\mathcal{F}_n(X; \mathbb{K})$ filtration, 119

$\Gamma$ Jacobi diagram, 42

$\Gamma_A$ subgraph of $\Gamma$, 202

$\widetilde{\operatorname{hol}}_\gamma(\eta_{B,A})$ A-holonomy, 378

$\widetilde{\operatorname{hol}}_{[a,b]}(.)$ A-holonomy, 382

IHX, 129

Integrals over configuration spaces

$I(\Gamma, A)$, 203, 208

$I(R, L, \Gamma, o(\Gamma), (\omega(i))_{i \in \underline{3n}})$, 148

$I(C, L, \Gamma, o(\Gamma), (\omega(i)))$, 270

$I_\theta(K, \tau)$ associated to $\xleftarrow{\phantom{=} \rightleftharpoons} \phantom{=} \xrightarrow{\phantom{=} \}$, 153

$I_\theta(K, \tau)$ for a long knot, 274

$\iota$ involution of $C_2(R)$, 75

$j_E: E(\Gamma) \hookrightarrow A$, 148

$K(V; f)$ kid set, 172
Lagrangians

$\mathcal{L}_A$, 46

$\mathcal{L}_A^{\mathbb{Z}/2\mathbb{Z}}$, 503, 505

$\lambda_{CW}$ Casson–Walker invariant, 43

$L(\mathcal{E})$ set of leaves, 308

$L(\mathcal{T})$ set of leaves, 308

$N_\epsilon(X)$ $\epsilon$-neighborhood, 246

$\omega_I$ propagating form, 483, 491

$\omega(p^i)$ 2-form, 490

$\omega(\gamma, \tau_b)$ 2-form, 507

Open configuration sets

$O(A, K(A), b, T)$, 184

Operation sets

$\mathcal{O}_{\mathcal{L}}^{\mathbb{Q}}$, 46

$\mathcal{O}_{\mathcal{L}}^{\mathbb{Z}}$, 46

Parenthesizations

$\mathcal{P}_X$, 325

$\mathcal{P}_x$, 325

$\mathcal{P}'_X$, 349

$\mathcal{P}'_X$, 350

$\mathcal{P}_{X,D}$, 349

$\mathcal{P}'_x$, 364

Projections

$p^c: \mathcal{A}(\emptyset) \to \mathcal{A}^c(\emptyset)$, 154

$\check{p}: \mathcal{A}(\mathcal{L}) \to \check{\mathcal{A}}(\mathcal{L})$, 154

$p_{S^2}$ to $S^2$, 42, 62

$p_\infty: S_\infty^2 \to S^2$, 69

$p_\tau: \partial C_2(R) \to S^2$, 72

$P_r(A)$ set of $r$-tuples, 378

$\mathcal{P}_{\ge 1}(V)$ set of nonempty subsets, 208

$\mathcal{P}_{\ge 2}(V)$, 198

$Q(v; \check{\Gamma})$ configuration space, 227

3-manifolds

$\check{R}$ punctured 3-manifold, 42

$\check{R}(\mathcal{C})$, 269

$R_{(K;p/q)}$ Dehn surgery, 461

$\rho_\theta$, 508

$\rho: B^3 \to SO(3)$, 80

$\tilde{\rho}: S^3 \to SO(3)$, 80

$S(T) = (T \setminus \{0\})/\mathbb{R}^{+*}$, 65

$S_H^2$ subset of $S^2$, 416

$\mathbb{S}$ unit sphere, 176

$\check{\mathcal{S}}_V(T)$ open configuration space, 177

$\bar{\mathcal{S}}_V(T)$ configuration space, 176

$\check{\mathcal{S}}(T_\infty R, A)$ open configuration space, 182

$T(\Gamma) = \{\text{trivalent vert. of } \Gamma\}$, 42

$\mathcal{T}_k$ twist, 508

$\tau_s$ standard parallelization of $\mathbb{R}^3$, 72

$U(\Gamma) = \{\text{univalent vert. of } \Gamma\}$, 42

$U^{-}K$ unit negative tangent vectors to $K$, 152

$U^{+}K$ unit positive tangent vectors to $K$, 152

$V(\Gamma) = \{\text{vertices of } \Gamma\}$, 42

$\mathcal{V}(\Gamma)$ configuration space, 347

$\mathcal{V}(y, \Gamma)$ configuration space, 363

$X(\mathcal{T})$ singular space associated to a tree $\mathcal{T}$, 309

$\zeta_\Gamma$ averaging coefficient, 154, 160

$\mathcal{Z}$ and some variants (see also the summary in the next pages)

$z$, 222

$z(\check{R}, (\omega(i)))$, 154

$\check{z}$, 223

$Z_n(\check{R}, L, (\omega(i)))$, 154

$Z_{n,A}(\check{R}, L, (\omega(i)))$, 160

$Z_n(\mathcal{C}, L, (\omega(i)))$, 274

$Z(\check{R}, L, \tau)$, 157

$Z(\mathcal{C}, L, \tau)$, 275
$\mathcal{Z}(R, L)$, 157

$\mathcal{Z}(R)$, 157

$\mathcal{Z}(C, L)$, 275

$\check{\mathcal{Z}}$, 157, 447

$\mathcal{Z}^f(\check{R}, L, L_\|)$, 166

$\mathcal{Z}^f(C, (L, L_\|))$, 277

$\overline{\mathcal{Z}}$ mod 1T, 448

$\check{\mathcal{Z}}$, 448
<figure><img src="image_172.png" /></figure>
# Summarizing the main definitions of Z

In the informal summary below, we first recall the various definitions of Z and its variants ($\tilde{Z}$, $z$, $\mathcal{Z}^f$) for a given link embedding $L = \bigsqcup_{j=1}^k K_j$ into a $\mathbb{Q}$-sphere $R$. The punctured $\tilde{R} = R \setminus \{\infty\}$ is equipped with a parallelization $\tau$ that makes $(\tilde{R}, \tau)$ an asymptotic rational homology $\mathbb{R}^3$ as in Definition 3.8.

For $n \in \mathbb{N}$, for a family $(\omega(i))_{i \in \underline{3n}}$ of propagating forms of $C_2(R)$ as in Section 3.3, we have

$$Z_n(\tilde{R}, L, (\omega(i))_{i \in \underline{3n}}) \stackrel{\text{Not. 7.16}}{=} \sum_{\Gamma \in \mathcal{D}_n^e(\mathcal{L})} \zeta_{\Gamma} I(R, L, \Gamma, (\omega(i))_{i \in \underline{3n}}) [\Gamma] \in \mathcal{A}_n(\mathcal{L}).$$

If the $\omega(i)$ are propagating forms of $(C_2(R), \tau)$ as in Definition 3.9, and if they are homogeneous as in Definition 3.13, then we have

$$Z_n(\tilde{R}, L, \tau) \stackrel{\text{Thm. 7.20}}{=} Z_n(\tilde{R}, L, (\omega(i))_{i \in \underline{3n}})$$

and

$$Z(\tilde{R}, L, \tau) = (Z_n(\tilde{R}, L, \tau))_{n \in \mathbb{N}} = \sum_{n \in \mathbb{N}} Z_n(\tilde{R}, L, \tau).$$

With the anomalies $\alpha$ of Section 10.3 and $\beta$ of Section 10.2, the Pontragin number $p_1(\tau)$ of Definition 5.13, and the definition of $I_\theta$ in Lemma 7.15, we have

$$\mathcal{Z}(R, L) \stackrel{\text{Thm. 7.20}}{=} \exp\left(-\frac{1}{4}p_1(\tau)\beta\right) \prod_{j=1}^k \left(\exp(-I_\theta(K_j, \tau)\alpha)\#_j\right) Z(\tilde{R}, L, \tau),$$

<footnote n="1">¹When $\omega(i) = \omega$ for all $i$, we set $Z_n(\tilde{R}, L, \omega) = Z_n(\tilde{R}, L, (\omega(i))_{i \in \underline{3n}})$ as in Proposition 7.25. The use of a family of distinct $\omega(i)$ allows us to discretize the definition of $\mathcal{Z}$ in Chapter 11, thanks to Version 7.40 of Theorem 7.20.</footnote>

<footnote n="2">²The similar letters $Z$ and $\mathcal{Z}$ respectively denote a function involving auxiliary data and the induced invariant.</footnote>
and

$$ \mathcal{Z}(R) = \mathcal{Z}(R, \emptyset). $$

With the projection $p^c: \mathcal{A}(\emptyset) \to \mathcal{A}^c(\emptyset)$ on the connected part of Notation 7.16, we have

$$ z(R) = p^c(\mathcal{Z}(R)) \text{ and } \mathcal{Z}(R) = \exp(z(R)). $$

With the projection $\tilde{p}: \mathcal{A}(\mathcal{L}) \to \tilde{\mathcal{A}}(\mathcal{L})$ of Notation 7.16, which forgets diagrams without univalent vertices, we have

$$ \tilde{\mathcal{Z}} = \tilde{p} \circ \mathcal{Z} \text{ and } \mathcal{Z}(R, L) = \mathcal{Z}(R)\tilde{\mathcal{Z}}(R, L). $$

When $L = \bigsqcup_{j=1}^k K_j$ has a parallel $L_\parallel = \bigsqcup_{j=1}^k K_{j\parallel}$, the framed version $\mathcal{Z}^f$ of $\mathcal{Z}$ satisfies

$$ \mathcal{Z}^f(\tilde{R}, L, L_\parallel) \stackrel{\text{Def. 7.41}}{=} \prod_{j=1}^k \left( \exp(lk(K_j, K_{j\parallel})\alpha) \#_j \right) \mathcal{Z}(R, L). $$

Now, we give a similar summary for the extension of $\mathcal{Z}$ to tangles in $\mathbb{Q}$-cylinders of the book's third part. We fix such a $\mathbb{Q}$-cylinder $\mathcal{C}$, the associated $\mathbb{Q}$-sphere $R(\mathcal{C})$, a tangle $L$ in $\mathcal{C}$, and the associated long tangle also denoted by $L$ in $\tilde{R}(\mathcal{C})$ as in Definition 12.1. Theorem 12.7 defines $Z(\mathcal{C}, L, \tau)$ (resp. $Z_n(\mathcal{C}, L, (\omega(i))_{i \in \mathcal{3n}})$) to be a natural extension of $Z(R(\mathcal{C}), L, \tau)$ (resp. $Z_n(R(\mathcal{C}), L, (\omega(i))_{i \in \mathcal{3n}})$) from the case where $L$ is a link in $\mathcal{C}$ to the case where $L$ is any tangle of $\mathcal{C}$ canonically extended to a long tangle as in Definition 12.1. We have

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

with respect to Definition 12.6 for $I_\theta(K_j, \tau)$ when $K_j$ is a long component. Definition 12.12 of

$$ \mathcal{Z}^f(\mathcal{C}, (L, L_\parallel)) = \prod_{j=1}^{k} \left( \exp(lk(K_j, K_{j\parallel})\alpha) \#_j \right) \mathcal{Z}(\mathcal{C}, L). $$

extends from tangles to $q$-tangles, which are cobordisms between limit configurations, by a natural limit process as in Remark 13.11.

Again, discrete computations and proofs of properties involve straight tangles as in Section 16.3 and distinct propagating forms dual to propagating chains as in Chapter 11. Theorems 16.9 and 16.16, and Definition 16.44 describe variants of $\mathcal{Z}$ associated to such data. These variants depend on 2-forms over $S^2$ in a way described in Theorem 16.45.