# Invariants of links and 3-manifolds from graph configurations

Christine Lescop

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Christine Lescop. Invariants of links and 3-manifolds from graph configurations. 2020. hal-02465054

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Invariants of links and 3-manifolds from graph configurations

<figure><img src="image_2.png" /></figure>

Christine Lescop<fnref n="1" />

September 17, 2024

<footnote n="1">¹Institut Fourier, CNRS, Université Grenoble Alpes, 38000 Grenoble, France</footnote>
<figure><img src="image_3.png" /></figure>
<table><tr><td>13 More on the functor Z<sup>f</sup></td><td>287</td></tr><tr><td>13.1 Tangles and q-tangles</td><td>287</td></tr><tr><td>13.2 Definition of Z<sup>f</sup> for all q-tangles</td><td>294</td></tr><tr><td>13.3 Properties of the functor Z<sup>f</sup></td><td>298</td></tr></table>

<table><tr><td>14 Invariance of Z<sup>f</sup> for long tangles</td><td>307</td></tr><tr><td>14.1 Singular models associated to trees</td><td>307</td></tr><tr><td>14.2 Configuration spaces of graphs on long tangles</td><td>321</td></tr><tr><td>14.3 Variations of integrals on configuration spaces of long tangles</td><td>342</td></tr></table>

<table><tr><td>15 The invariant Z as a holonomy for braids</td><td>347</td></tr><tr><td>15.1 On the structure of V(Γ)</td><td>347</td></tr><tr><td>15.2 A one-form on S&#773;<sub>B</sub>(C)</td><td>366</td></tr></table>

<table><tr><td>16 Discretizable variants of Z<sup>f</sup> and extensions to q-tangles</td><td>377</td></tr><tr><td>16.1 Discretizable holonomies</td><td>377</td></tr><tr><td>16.2 Variants of Z<sup>f</sup> for tangles</td><td>383</td></tr><tr><td>16.3 Straight tangles</td><td>386</td></tr><tr><td>16.4 Semi-algebraic structures on some configuration spaces</td><td>394</td></tr><tr><td>16.5 Extending Z<sup>f</sup> to q-tangles</td><td>398</td></tr></table>

<table><tr><td>17 Justifying the properties of Z<sup>f</sup></td><td>415</td></tr><tr><td>17.1 Transversality and rationality</td><td>416</td></tr><tr><td>17.2 Functionality</td><td>422</td></tr><tr><td>17.3 Insertion of a tangle in a trivial q-braid.</td><td>428</td></tr><tr><td>17.4 Duplication property</td><td>431</td></tr><tr><td>17.5 Behavior of Z<sup>f</sup> with respect to the coproduct</td><td>446</td></tr><tr><td>17.6 A proof of universality</td><td>448</td></tr></table>

IV Universality 455

<table><tr><td>18 The main universality statements and their corollaries</td><td>457</td></tr><tr><td>18.1 Universality with respect to Lagrangian-preserving surgeries</td><td>457</td></tr><tr><td>18.2 On Dehn surgeries</td><td>460</td></tr><tr><td>18.3 Direct proof of a surgery formula for Θ</td><td>464</td></tr><tr><td>18.3.1 A Lagrangian-preserving surgery associated to a Seifert surface</td><td>465</td></tr><tr><td>18.3.2 A direct proof of the Casson surgery formula</td><td>470</td></tr><tr><td>18.4 Finite type invariants of Z-spheres</td><td>472</td></tr><tr><td>18.5 Finite type invariants of Q-spheres</td><td>476</td></tr><tr><td>18.6 Identifying Θ with the Casson-Walker invariant</td><td>479</td></tr><tr><td>18.7 Sketch of the proof of Theorem 18.6</td><td>480</td></tr></table>
<table><tbody><tr><td>18.8</td><td>Mixed universality statements</td><td>488</td></tr><tr><td>19</td><td>More flexible definitions of Z using pseudo-parallelizations</td><td>501</td></tr><tr><td>19.1</td><td>Why we need pseudo-parallelizations</td><td>501</td></tr><tr><td>19.2</td><td>Definition of pseudo-parallelizations</td><td>506</td></tr><tr><td>19.3</td><td>Integration of homogeneous propagating forms along surfaces</td><td>510</td></tr><tr><td>19.4</td><td>Anomalous terms for pseudo-parallelizations</td><td>514</td></tr><tr><td>19.5</td><td>Proof of Theorem 19.17</td><td>526</td></tr><tr><td>20</td><td>Simultaneous normalization of propagating forms</td><td>531</td></tr><tr><td>20.1</td><td>Sketch</td><td>531</td></tr><tr><td>20.2</td><td>Proof of Proposition 20.1</td><td>535</td></tr><tr><td>20.3</td><td>Proof of Lemma 20.2</td><td>539</td></tr><tr><td>21</td><td>Much more flexible definitions of Z</td><td>549</td></tr><tr><td>21.1</td><td>More propagating forms associated to pseudo-parallelizations</td><td>549</td></tr><tr><td>21.2</td><td>Pseudo-sections associated with pseudo-parallelizations</td><td>556</td></tr><tr><td>21.3</td><td>Definition of Z with respect to pseudo-sections</td><td>560</td></tr><tr><td>A</td><td>Some basic algebraic topology</td><td>563</td></tr><tr><td>A.1</td><td>Homology</td><td>563</td></tr><tr><td>A.2</td><td>Homotopy groups</td><td>572</td></tr><tr><td>B</td><td>Differential forms and de Rham cohomology</td><td>575</td></tr><tr><td>B.1</td><td>Differential forms</td><td>575</td></tr><tr><td>B.2</td><td>De Rham cohomology</td><td>577</td></tr><tr><td colspan="2">Bibliography</td><td>583</td></tr><tr><td colspan="2">Terminology</td><td>595</td></tr><tr><td colspan="2">Index of notation</td><td>599</td></tr><tr><td colspan="2">Summarizing the definitions of Z</td><td>603</td></tr></tbody></table>

**Keywords:** Knots, 3-manifolds, finite type invariants, homology 3-spheres, linking number, Theta invariant, Casson–Walker invariant, Jacobi diagrams, perturbative expansion of Chern–Simons theory, configuration space integrals, parallelizations of 3-manifolds, first Pontrjagin class

**MSC 2020:** 57K16 57K31 57K30 55R80 57R20 81Q30
À Lucien Guillou,
À nos 26 ans de bonheur ensemble,
À nos deux enfants, Gaëlle et Ronan.
# Preface

In this book, we explain how to count graph configurations to obtain invariants for 3-manifolds and knots in these 3-manifolds, and we investigate the properties of the obtained invariants.

The simplest of these invariants is the linking number of two disjoint knots in the ambient space $\mathbb{R}^3$. Gauss defined it in 1833 [Gau77]. As we review in Section 1.2, this linking number counts configurations

<figure><img src="image_4.png" /></figure>

as the degree of an associated Gauss map.

Many mysterious knot invariants called “quantum invariants” were introduced in the mid-80s, starting with the Jones polynomial. Witten explained how to obtain many of them from the perturbative expansion of the Chern-Simons theory in a seminal article [Wit89]. This physicist viewpoint led Guadagnini, Martellini, Mintchev [GMM90] and Bar-Natan [BN95b] to show in what sense a coefficient $w_2$ of the Jones polynomial counts configurations of the graphs

<figure><img src="image_5.png" /></figure>

The theory of Vassiliev invariants reviewed in Chapter 6 associates a degree in $(\mathbb{N} \cup \{\infty\})$ to a numerical knot invariant. The only knot invariants of degree 0 are the constant functions. The knot invariants of degree 2 are linear combinations of $w_2$ and the constant function that maps every knot to 1. The Jones polynomial can be renormalized into a series whose coefficients are
finite-degree knot invariants. Altschüler and Freidel showed that every degree $n$ real-valued knot invariant may be obtained by “counting” configurations of graphs with at most $2n$ vertices as explained in this book [AF97]. The knot invariants counting graph configurations mentioned above are assembled in a *universal Vassiliev invariant* $\mathcal{Z}(S^3, .)$ valued in a product of vector spaces generated by some unitivalent graphs called *Jacobi diagrams*. Kontsevich had constructed another universal Vassiliev invariant with similar properties called *the Kontsevich integral* [Kon93, BN95a]. The Kontsevich integral $Z_K$ may be defined combinatorially from planar knot diagrams. It has been extensively studied. To my knowledge, the coincidence of the spatial invariant $\mathcal{Z}(S^3, .)$ with $Z_K$ is an open problem.

Developing the Witten approach further, Kontsevich outlined a way to count trivalent graphs in more general 3-manifolds and define a topological graded invariant $\mathcal{Z}$ for them [Kon94]. These more general manifolds are the 3-dimensional $\mathbb{Q}$-spheres, simply called $\mathbb{Q}$-spheres in this book. They are the closed 3-manifolds with the same rational homology as the standard 3-dimensional sphere $S^3$. They include the 3-manifolds with the same $\mathbb{Z}$-homology as $S^3$ called $\mathbb{Z}$-spheres. A $\mathbb{Z}$-sphere is a closed orientable three-manifold in which knots bound orientable compact surfaces. The degree one part $\mathcal{Z}_1$ of the Kontsevich invariant $\mathcal{Z}$ of $\mathbb{Q}$-spheres is determined by a real-valued invariant $\Theta$ of $\mathbb{Q}$-spheres, which counts configurations of the graph $\Theta$ in the manifold.

Ohtsuki, Habiro, Goussarov, and others developed theories of finite type invariants of $\mathbb{Z}$-spheres analogous to the Vassiliev theory for knots in $\mathbb{R}^3$ [Oht96, GGP01]. Kuperberg and Thurston showed why the Kontsevich invariant $\mathcal{Z}$ of $\mathbb{Z}$-spheres obtained by counting graphs in these manifolds is also *universal* with respect to the above theories [KT99]. This universality result implies that any real-valued invariant of $\mathbb{Z}$-spheres of degree $2n$ (with respect to one of the equivalent developed theories) is a combination of invariants counting configurations of graphs with at most $2n$ vertices.

In 1985, Casson had defined an invariant of $\mathbb{Z}$-spheres. The Casson invariant “counts” conjugacy classes of irreducible $SU(2)$-representations of the $\mathbb{Z}$-spheres fundamental groups. Their universality result allowed Kuperberg and Thurston to show that $\Theta$ and the Casson invariant are proportional. In particular, the Casson invariant also “counts” configurations of the graph $\Theta$. For a $\mathbb{Z}$-sphere $R$ equipped with a basepoint $\infty$, the configurations are counted in a suitable compactification $C_2(R)$ of the space of ordered pairs of distinct points in the punctured manifold $(\tilde{R} = R \setminus \{\infty\})$. The set of counted configurations is the intersection in $C_2(R)$ of three transverse codimension-2 submanifolds called *propagating chains*, and $\Theta(R)$ is their algebraic intersection number. Dually, the invariant $\Theta(R)$ is the integral over $C_2(R)$ of the
cube of a *propagating* closed 2-form. Propagating chains and propagating
forms both represent the linking form on $R$. We call them *propagators*. They
are the main ingredient used to count graph configurations in this book.
They are associated with the graph edges. They are precisely defined in
Chapter 3. When $R$ is $\mathbb{Z}$-sphere, results of Pontrjagin and Rohlin in the
1950s [Roh52] imply that the punctured $\tilde{R}$ can be equipped with a preferred
homotopy class of parallelizations. For a general $\mathbb{Q}$-sphere, the invariant $\Theta$
is first introduced as an invariant of a pair $(R, \tau)$, where $\tau$ is a parallelization
of $\tilde{R}$. It is next corrected with the help of a relative first Pontrjagin class to
become an invariant of $R$. Chapter 4 contains the complete construction of
$\Theta$, and Chapter 5 establishes the needed properties of Pontrjagin classes.

Kuperberg and I associated explicit propagating chains to Morse func-
tions and associated Morse flows. These propagators reviewed in Section 1.2.8
allowed me to express the Theta invariant in terms of Heegaard diagrams
[Les15a]. With this type of propagator, the “counted” graph configurations
either map an edge of the graph into a flow line, or map the edge ends into
descending manifolds or ascending manifolds of critical points of the Morse
function. Fukaya proposed such a way of counting graphs [Fuk96]. Many
authors, including Watanabe and Shimizu, further studied it.

In the book's second part, we define and study an invariant $\mathcal{Z}(R, L)$ for
a link $L$ in a $\mathbb{Q}$-sphere $R$. This invariant generalizes both $\mathcal{Z}(S^3, L)$ and
$\mathcal{Z}(R) = \mathcal{Z}(R, \emptyset)$. Our definitions are more flexible than the original ones.
We prove generalizations of the mentioned universality results in the book's
fourth part.

To get more properties of $\mathcal{Z}$, we cut our pairs $(R, L)$ of links $L$ in $\mathbb{Q}$-spheres $R$ into pieces called *tangles in $\mathbb{Q}$-cylinders*. These pieces can be composed in various ways. In the book's third part, we generalize $\mathcal{Z}$ to a functorial invariant of framed tangles in $\mathbb{Q}$-cylinders, and we prove that it behaves well under the various allowed compositions.

Our first chapter is a more complete and much longer preface to this book.
It contains several introductions. Section 1.1 is a short summary for experts.
Other readers can start with Section 1.2, a slow informal introduction based
on examples from which a broad audience can get the flavor of the studied
topics and hopefully become interested. Section 1.3 is an independent, more
formal, mathematical overview of the contents. It is accessible to beginners
in topology after the warm-up of Section 1.2. Section 1.4 describes the book
organization. Section 1.5 outlines why I wrote this book and what I consider
original and new.

Apart from this preface and the first chapter, which has some parts written for experts and is sometimes imprecise, the rest of the book is precise, detailed, and mostly self-contained. The only prerequisites are basic notions
of algebraic topology and de Rham cohomology, surveyed in the appendices.

In 2018, Watanabe disproved a long-standing conjecture called the 4-dimensional Smale conjecture by constructing a topologically trivial $S^4$-bundle over $S^2$, which is not smoothly equivalent to the trivial bundle $S^4 \times S^2$ [Wat18b]. He distinguished his exotic $S^4 \times S^2$ from the standard $S^4 \times S^2$ using characteristic classes introduced by Kontsevich [Kon94]. The involved Kontsevich-Watanabe characteristic class of a smooth topologically trivial $S^4$-bundle over $S^2$ counts configurations of the complete graph $\Delta$ with four vertices in the total space of the bundle. The ideas and techniques used by Watanabe are similar to those presented in this book. Even though we only count graph configurations in dimension 3, this book can also serve as an introduction to the work of Watanabe and other articles about invariants counting graph configurations in higher dimensions.

I thank the referees for their careful reading and their helpful comments.
# Part I

## Introduction
<figure><img src="image_6.png" /></figure>
# Chapter 1

## Introductions

In this chapter, we propose several introductions to this book:

• a short abstract for experts in Section 1.1,

• a slow informal introduction based on examples in Section 1.2, from which a broad audience can get the flavor of the studied topics and hopefully become interested,

• an independent, more formal, mathematical overview of the contents, to which the experts can go directly, in Section 1.3, and

• a section on the genesis of this book in Section 1.5.

I apologize for some repetitions due to this structure. Section 1.4 describes the book organization. The chapter ends with a list of open problems in Section 1.6

Unlike this first chapter, which has some parts written for experts and is sometimes imprecise, the rest of the book is precise, detailed, and mostly self-contained. It relies only on the basic notions of algebraic topology and the basic notions of de Rham cohomology, surveyed in the appendices.

### 1.1 An abstract for experts

**Very first conventions.** Unless otherwise mentioned, manifolds are smooth, but they may have boundary and corners. Let $\mathbb{K}$ be $\mathbb{Z}$ or $\mathbb{Q}$. In this book, a $\mathbb{K}$-sphere is a compact oriented 3-dimensional manifold with the same topology with coefficients in $\mathbb{K}$ as the standard unit sphere $S^3$ of $\mathbb{R}^4$. The unit disk of $\mathbb{C}$ is denoted by $D_1$. A $\mathbb{K}$-ball (resp. a $\mathbb{K}$-cylinder, a *genus g K-handlebody*) is a compact oriented 3-dimensional manifold $A$ with the same
homology with coefficients in $\mathbb{K}$ as the standard unit ball $B^3$ of $\mathbb{R}^3$, (resp. the cylinder $D_1 \times [0, 1]$, the standard solid handlebody $H_g$ of Figure 1.1), such that a neighborhood of the boundary of $A$ (which is necessarily homeomorphic to the boundary of its model-ball, cylinder or handlebody) is identified with a neighborhood of the boundary of its model by a smooth diffeomorphism.

<figure><img src="image_7.png" /><figcaption>Figure 1.1: The genus $g$ handlebody $H_g$</figcaption></figure>

In “Q-spheres”, $\mathbb{Q}$ is a shortcut for *rational homology*. So $\mathbb{Q}$-spheres are also called *rational homology spheres* or *rational homology 3-spheres*, while $\mathbb{Z}$-spheres are also called *integer homology 3-spheres*.

**Abstract.** In this book, following Edward Witten [Wit89], Maxim Kontsevich [Kon94], Greg Kuperberg and Dylan Thurston [KT99], we define an invariant $\mathcal{Z}$ of $n$-component links $L$ in rational homology 3-spheres $R$, and we study its properties. The invariant $\mathcal{Z}$ is often called “the perturbative expansion of the Chern–Simons theory”. It is valued in a graded space $\mathcal{A}(\bigsqcup_{j=1}^k S^1)$ generated by Jacobi diagrams $\Gamma$ on $\bigsqcup_{j=1}^k S^1$. These diagrams are a special kind of Feynman diagrams. They are uni-trivalent. The invariant $\mathcal{Z}(L)$ is a combination $\mathcal{Z}(L) = \sum_{\Gamma} \mathcal{Z}_{\Gamma}(L)[\Gamma]$ for coefficients $\mathcal{Z}_{\Gamma}(L)$ that “count” embeddings of $\Gamma$ in $R$ mapping the univalent vertices of $\Gamma$ to $L$, in a sense explained in the book. The coefficients $\mathcal{Z}_{\Gamma}(L)$ may be defined as integrals over configuration spaces or, in a dual way, as algebraic intersections in the same configuration spaces.

When $R = S^3$, the invariant $\mathcal{Z}$ is a universal Vassiliev link invariant studied by many authors, including Enore Guadagnini, Maurizio Martellini, and Mihail Mintchev [GMM90], Dror Bar-Natan [BN95b], Raoul Bott and Clifford Taubes [BT94], Daniel Altschüler and Laurent Freidel [AF97], Dylan Thurston [Thu99], and Sylvain Poirier [Poi02]. . . This book contains a more flexible definition of this invariant.

*Rational LP-surgeries* in $\mathbb{Q}$-spheres are replacements of rational homology handlebodies by other rational homology handlebodies, in a way that does not change the linking number of curves outside the replaced handlebodies. We prove that the restriction of $\mathcal{Z}$ to $\mathbb{Q}$-spheres (equipped with
empty links) is a universal finite type invariant with respect to rational LP-surgeries. Together with recent results of Gwénaël Massuyeau [Mas14] and Delphine Moussard [Mou12], this implies that the restriction of $\mathcal{Z}$ to $\mathbb{Q}$-spheres contains the same information as the Lê–Murakami–Ohtsuki LMO invariant [LMO98] for these manifolds. This also implies that the degree one part of $\mathcal{Z}$ is the Casson–Walker invariant.

We extend $\mathcal{Z}$ to a functorial invariant of framed tangles in rational homology cylinders. We describe the behavior of this functor under various operations including some cabling operations. We also compute iterated derivatives of this extended invariant with respect to discrete derivatives associated to the main theories of finite type invariants.

## 1.2 A slow informal introduction for beginners

In this introduction, we describe the contents of this book to a broad audience, including graduate students. We start with examples to give a flavor of the topics studied in this book and to introduce ideas, conventions, and methods used later.

This book is about invariants of links and 3-manifolds that count graph configurations. The first example of such an invariant goes back to Carl Friedrich Gauss in 1833 [Gau77]. It is the linking number of two knots. We discuss it in Subsection 1.2.1.

At the end of the eighties, Edward Witten's insight into the perturbative expansion of the Chern–Simons theory [Wit89] gave birth to trickier examples. Following Witten's ideas, Enore Guadagnini, Maurizio Martellini, and Mihail Mintchev [GMM90], and Dror Bar-Natan [BN95b] defined and studied another knot invariant $w_2$, which counts configurations of some uni-trivalent graphs with 4 vertices. Maxim Kontsevich [Kon94], Raoul Bott and Clifford Taubes [BT94] [Bot96], Daniel Altschüler and Laurent Freidel [AF97], and others revisited and generalized the definition of $w_2$. In Subsection 1.2.5, we detail the example of $w_2$.

### 1.2.1 The linking number as a degree

Let $S^1$ denote the unit circle of the complex plane $\mathbb{C}$. We use “:” as a symbol for “such that”. So, the circle $S^1$ is the set $\{z : z \in \mathbb{C}, |z| = 1\}$. Consider a $\mathbb{C}^\infty$ embedding

$$J \sqcup K : S^1 \sqcup S^1 \hookrightarrow \mathbb{R}^3$$
of the disjoint union $S^1 \sqcup S^1$ of two circles in the ambient space $\mathbb{R}^3$, as the
example pictured in Figure 1.2. Such an embedding represents a 2-component
link.

<figure><img src="image_8.png" /><figcaption>Figure 1.2: A 2-component link in $\mathbb{R}^3$</figcaption></figure>

It induces the Gauss map

<figure><img src="image_9.png" /></figure>

**Definition 1.1.** The Gauss linking number $lk_G(J, K)$ of the disjoint knot embeddings $J$ and $K$ is the degree of the Gauss map $p_{JK}$.

Below, we give our favorite definition of the degree for this book. To do
so, we first agree on conventions used throughout the book.

1.2.2 On orientations of manifolds and degrees of maps

We work with smooth (i.e., $C^\infty$) manifolds locally diffeomorphic to open
subspaces of $[0, 1]^n$. These manifolds are precisely described in Sections 2.1.1
and 2.1.4. They may have boundaries and ridges (or corners). The cube
$[0, 1]^3$ is an example of such a manifold. Its edges and its vertices are *ridges*.

**Conventions 1.2.** Let $M$ be such a manifold. The *interior* of $M$ consists of the points with a neighborhood diffeomorphic to an open subspace of $\mathbb{R}^n$. The *boundary* of $M$ is the complement of its interior in $M$.

An *orientation* of a real vector space *V* of positive dimension is a basis
of *V* up to a change of basis with positive determinant. When *V* = {0},
an orientation of *V* is an element of {-1, 1}. An *orientation* of a smooth
*n*-manifold is an orientation of its tangent space at each point of its interior,
defined in a continuous way. (A local diffeomorphism $h$ of $\mathbb{R}^n$ is *orientation-preserving* at $x$ if and only if the Jacobian determinant of its derivative $T_x h$ is positive. If the transition maps $\phi_j \circ \phi_i^{-1}$ of an *atlas* $(\phi_i)_{i \in I}$ of a manifold $M$ (as in Subsection 2.1.1) are orientation-preserving (at every point) for $\{i, j\} \subseteq I$, then the manifold $M$ is *oriented* by this atlas.) Unless otherwise mentioned, manifolds are smooth, oriented, compact, and considered up to orientation-preserving diffeomorphisms, in this book. Products are oriented by the order of the factors. More generally, unless otherwise mentioned, the order of appearance of coordinates or parameters orients manifolds. When $M$ is an oriented manifold, $(-M)$ denotes the same manifold, equipped with the opposite orientation.

The boundary $\partial M$ of an oriented manifold $M$ is oriented by the *outward normal first convention*: If $x \in \partial M$ is not in a ridge, then the outward normal to $M$ at $x$ followed by an oriented basis of $T_x \partial M$ induce the orientation of $M$.

For example, the standard orientation of the disk in the plane induces the traditional counterclockwise orientation of the circle, as the following picture shows.

<figure><img src="image_10.png" /></figure>

As another example, the sphere $S^2$ is oriented as the boundary of the unit ball $B^3$ of $\mathbb{R}^3$, which has the standard orientation induced by (thumb, index finger (2), middle finger (3)) of the right hand.

<figure><img src="image_11.png" /></figure>

**Definitions 1.3.** Let $M$ and $N$ be smooth manifolds, and let $p: M \to N$ be a smooth map from $M$ to $N$. Assume that the boundary of $M$ is empty. A point $x$ of $M$ is called a *regular point of p* if the tangent map

$$T_x p: T_x M \to T_y N$$

at $x$ is surjective. A point $y$ of $N$ is called a *regular value of p* if $p^{-1}(y)$ contains only regular points.
Our $d$-manifolds are covered by countably many open sets diffeomorphic to open subsets of $[0, 1]^d$. The following *Morse–Sard theorem* is proved in the book [Hir94] by Morris W. Hirsch. See [Hir94, Chapter 3, Theorem 1.3, p. 69].

**Theorem 1.4 (Morse–Sard theorem).** *The set of regular values of a smooth map from a manifold to another such is dense.*<fnref n="1" />

If the source $M$ of a smooth map $p: M \to N$ is compact, then the set of its regular values is furthermore open. In general, the source may have boundary and ridges. Then a point $y$ is a *regular value* of $p$, if $y$ is a regular value of the restrictions of $p$ to the interior of $M$ and to all the open, smooth faces (or strata) of $M$ (of any codimension). In particular, if the dimensions of $M$ and $N$ coincide, then a regular value is not in the image of the boundary $\partial M$.

**Definition 1.5.** Assume that $M$ and $N$ are oriented, $M$ is compact, and the dimension of $M$ coincides with the dimension of $N$. Then the (*differential*) degree of $p$ at a regular value $y$ of $N$ is the (finite) sum running over the $x \in p^{-1}(y)$ of the signs of the determinants of $T_x p$ (in oriented charts). In this case, the differential degree of $p$ extends as a continuous function $\text{deg}(p)$ from $N \setminus p(\partial M)$ to $\mathbb{Z}$, as we prove in detail in Lemma 2.3. In particular, if the boundary of $M$ is empty and if $N$ is connected, then $\text{deg}(p)$ extends as a constant map from $N$ to $\mathbb{Z}$, whose value is called the *degree* of $p$. See [Mil97, Chapter 5].

Figure 1.3 shows the values of $\text{deg}(p)$ for the pictured vertical projection $p$ from the interval $[0, 1]$ to $\mathbb{R}$.

<figure><img src="image_12.png" /><figcaption>Figure 1.3: Degree of the vertical projection</figcaption></figure>

Another easy example in higher dimensions is the case of an orientation-
preserving embedding $p$. In this case, $\text{deg}(p)$ is 1 on the image of the interior
of $M$ and 0 outside the image of $M$.

<footnote n="1">¹It is even *residual*, i.e., it contains the intersection of a countable family of dense open sets.</footnote>
### 1.2.3 Back to the linking number

The Gauss linking number $lk_G(J, K)$ can be computed from a link diagram as in Figure 1.2 as follows. It is the differential degree of $p_{JK}$ at the vector $Y$ that points towards us. The set $p_{JK}^{-1}(Y)$ is the set of pairs $(w, z)$ of points for which the projections of $J(w)$ and $K(z)$ coincide, and $J(w)$ is under $K(z)$. They correspond to the crossings $J \nsim K$ and $K \nsim J$ of the diagram.

In a diagram, a crossing is *positive* if we turn counterclockwise from the arrow at the end of the upper strand towards the arrow at the end of the lower strand like $\nsim$. Otherwise, it is *negative* like $\nsim$.

Consider a positive crossing $J \nsim K$. Moving $J(w)$ along $J$ following the orientation of $J$ moves $p_{JK}(w, z)$ towards the southeast direction $Tpdw$. In the meantime, moving $K(z)$ along $K$ following the orientation of $K$ moves $p_{JK}(w, z)$ towards the northeast direction $Tpdz$. So the local orientation $\langle_{Tpdw}^{Tpdz}$ induced by the image of $p_{JK}$ around $Y \in S^2$ is $\langle_1^2$. Therefore, the contribution of a positive crossing to the degree is 1. It is easy to deduce that the contribution of a negative crossing is $(-1)$. We denote the cardinality of a set $A$ by $|A|$.

In particular, $|J \nsim K|$ is the number of occurrences of $J \nsim K$ in the diagram. We have thus proved that

$$ \deg_Y(p_{JK}) = |J \nsim K| - |K \nsim J|. $$

So we have

$$ lk_G(J, K) = |J \nsim K| - |K \nsim J|. $$

We similarly obtain $\deg_{-Y}(p_{JK}) = |K \nsim J| - |J \nsim K|$. This implies

$$ \begin{aligned} lk_G(J, K) &= |K \nsim J| - |J \nsim K| \\ &= \frac{1}{2} \left( |J \nsim K| + |K \nsim J| - |K \nsim J| - |J \nsim K| \right) \end{aligned} $$

and $lk_G(J, K) = lk_G(K, J)$.

In the example of Figure 1.2, the Gauss linking number $lk_G(J, K)$ is 2. For the *positive Hopf link* of Figure 1.4, we have $lk_G(J, K) = 1$. The Gauss linking number of the components of the *negative Hopf link* is equal to $-1$. It is zero for the *Whitehead link*.

Let $\omega_S$ be a 2-form on $S^2$. The integral of the pull-back $p_{JK}^*(\omega_S)$ of $\omega_S$ over $S^1 \times S^1$ is the integral over $S^2$ of $\deg(p_{JK})\omega_S$. Since the differential degree $\deg(p_{JK})$ of the Gauss map $p_{JK}$ is constant and equal to $lk_G(J, K)$
<figure><img src="image_13.png" /><figcaption>Figure 1.4: The Hopf links and the Whitehead link</figcaption></figure>

on the set of regular values of $p_{JK}$, we have

$$ \int_{S^1 \times S^1} p_{JK}^*(\omega_S) = \int_{S^2} \deg(p_{JK}) \omega_S = lk_G(J, K) \int_{S^2} \omega_S. $$

Denote the standard area form of $S^2$ by $4\pi\omega_{S^2}$. So $\omega_{S^2}$ is the homogeneous volume form of $S^2$ such that $\int_{S^2} \omega_{S^2} = 1$. In 1833, Carl Friedrich Gauss defined the linking number of $J$ and $K$ as an integral [Gau77]. In modern notation, his definition may be written as

$$ lk_G(J, K) = \int_{S^1 \times S^1} p_{JK}^*(\omega_{S^2}). $$

### 1.2.4 A first noninvariant count of graph configurations

The above Gauss linking number counts the configurations of the graph $v_J \leftrightarrow v_K$, for which $v_J$ is on the knot $J$ (or more precisely on the image $J(S^1)$ of the knot embedding $J$), the vertex $v_K$ is on the knot $K$, and the edge from $v_J$ to $v_K$ is a straight segment with an arbitrary generic fixed direction $X$ in $S^2$. Here, generic means that $X$ is a regular value of $p_{JK}$.<fnref n="2" /> A configuration of $v_J \leftrightarrow v_K$ is an injection $c: \{v_J, v_K\} \hookrightarrow (\mathbb{R}^3)^2$ such that $c(v_J) = J(z_J)$ for some $z_J \in S^1$ and $c(v_K) = K(z_K)$ for some $z_K \in S^1$. The corresponding configuration space is parametrized by $(z_J, z_K) \in S^1 \times S^1$. It is diffeomorphic to $S^1 \times S^1$. The configurations such that the edge from $v_J$ to $v_K$ is a straight segment with direction $X$ in $S^2$ are in one-to-one correspondence with $p_{JK}^{-1}(X)$. The local degree of $p_{JK}$ equips each of these configurations with a sign. The “count” of configurations with direction $X$ is the sum of these signs, which is nothing but the degree of $p_{JK}$ at $X$. So any choice of a generic $X$ will give the same integral result, which will not be changed by a continuous deformation of our embedding among embeddings. The graph

<footnote n="2">²This is a generic condition thanks to the recalled Morse–Sard theorem 1.4.</footnote>
$v_J \leftrightarrow v_K$ will also be denoted by $J \stackrel{\text{ configuration}}{\leftrightarrow} K$. In this diagram, the dashed circles show to which component the vertices must map.

Let $K: S^1 \hookrightarrow \mathbb{R}^3$ be a smooth embedding of the circle into $\mathbb{R}^3$. Such an embedding is called a *knot embedding*. An isotopy between two knot embeddings $K$ and $K_1$ is a smooth map $\psi: [0, 1] \times S^1 \to \mathbb{R}^3$ such that the restriction $\psi(t, \cdot)$ of $\psi$ to $\{t\} \times S^1$ is a knot embedding for any $t \in [0, 1]$, $\psi(0, \cdot) = K$, and $\psi(1, \cdot) = K_1$. When there exists such an isotopy, the knot embeddings $K$ and $K_1$ are said to be *isotopic* or in the same *isotopy class*. A *knot* is an isotopy class of knot embeddings.

Let us now try to count the configurations

$$c: \{v_1, v_2\} \hookrightarrow (\mathbb{R}^3)^2$$

of the graph $v_1 \leftrightarrow v_2$, for which $c(v_1)$ and $c(v_2)$ are two distinct points on the (image of) the knot embedding $K$, and the edge from $v_1$ to $v_2$ is a straight segment with an arbitrary direction $X$ in $S^2$. The graph $v_1 \leftrightarrow v_2$ is also denoted by $\stackrel{\text{ configuration}}{\leftrightarrow}$. The associated configuration space is

$$\tilde{C}(K; \stackrel{\text{ configuration}}{\leftrightarrow}) = \{ (K(z), K(z \exp(2i\pi t))) : z \in S^1, t \in ]0, 1[ \}.$$

(In this book, open and half-open intervals are denoted with square brackets. For example, $]0, 1[$ denotes the open interval $\{x : 0 < x < 1\}$. Similarly, $]0, 1[$ denotes the half-open interval $\{x : 0 < x \le 1\}$.) The configuration space $\tilde{C}(K; \stackrel{\text{ configuration}}{\leftrightarrow})$ is naturally identified with the open annulus $S^1 \times ]0, 1[$. We have a *Gauss direction map*

$$G_K: \tilde{C}(K; \stackrel{\text{ configuration}}{\leftrightarrow}) \to S^2 \\ c \mapsto \frac{1}{\|K(z \exp(2i\pi t)) - K(z)\|} (K(z \exp(2i\pi t)) - K(z))$$

and the degree of $G_K$ at an element $X$ of $S^2$ makes sense as soon as $X$ is a regular value of $G_K$ whose preimage is finite.

<figure><img src="image_14.png" /><figcaption>Figure 1.5: A configuration of a segment on $K$</figcaption></figure>

The annulus $\tilde{C}(K; \stackrel{\text{ configuration}}{\leftrightarrow})$ can be compactified to the closed annulus $C(K; \stackrel{\text{ configuration}}{\leftrightarrow}) = S^1 \times [0, 1]$, to which $G_K$ extends smoothly. The extended $G_K$, still denoted
by $G_K$, maps $(z, 0) \in S^1 \times \{0\}$ to the direction of the tangent vector to $K$ at $z$. It maps $(z, 1) \in S^1 \times \{1\}$ to the opposite direction. The closed annulus $C(K; \text{↔})$ is an example of a smooth manifold whose boundary is

$$ \partial C(K; \text{↔}) = S^1 \times \{0\} \cup (-S^1 \times \{1\}). $$

The degree of $G_K$ is a continuous map from $S^2 \setminus G_K(\partial C(K; \text{↔}))$ to $\mathbb{Z}$. Let us compute it for the following embeddings of the trivial knot.

Let $O$ be an embedding of the circle to the horizontal plane.<fnref n="3" /> The image under $G_O$ of the whole annulus is in the horizontal great circle of $S^2$. The set of regular values of $G_O$ is the complement of this circle, and hence the degree of $G_O$ is zero on all this set.

Let $K_1$ and $K_{-1}$ be embeddings of $S^1$ such that

* the images of $K_1$ and $K_{-1}$ project to the horizontal plane as in Figure 1.6,

* they lie in the horizontal plane everywhere except in a small ball around where they cross over, and

* they lie in the union of two orthogonal planes.

<figure><img src="image_15.png" /><figcaption>Figure 1.6: Diagrams of the trivial knot</figcaption></figure>

The image of the boundary of $C(K_{\pm 1}; \text{↔}) = S^1 \times [0, 1]$ in $S^2$ lies in the union of the great circles of the two planes. More precisely, it lies in the union of the horizontal plane and two vertical arcs, as in the following figure.

<figure><img src="image_16.png" /></figure>

Therefore, when $K = K_\epsilon$, for $\epsilon = \pm 1$, the degree of $G_{K_\epsilon}$ (extends as a map, which) is constant on each side of our horizontal equator. Computing it at the North Pole $\vec{N}$ as in Subsection 1.2.3, we find that the degree of $G_{K_\epsilon}$ is $\epsilon$ on the Northern Hemisphere. We similarly compute the degree of $G_{K_\epsilon}$ on the Southern Hemisphere. It is also $\epsilon$.

<footnote n="3">³Here, “the” horizontal plane is the plane $\mathbb{C} \times \{0\}$ of $\mathbb{R}^3$ viewed as $\mathbb{C} \times \mathbb{R}$.</footnote>
For any embedding $K: S^1 \hookrightarrow \mathbb{R}^3$, define

$$I_{\theta}(K) = \int_{\tilde{C}(K; \to)} G_{K}^{*}(\omega_{S^{2}}).$$

The real number $I_{\theta}(K)$ is the integral of $\deg(G_K)\omega_{S^2}$ over $S^2$. It can be seen as the *algebraic area* of $G_K(C(K; \to))$. The above degree evaluation allows us to compute the integrals $I_{\theta}(O) = 0$, $I_{\theta}(K_1) = 1$, and $I_{\theta}(K_{-1}) = -1$.

More generally, say that a knot embedding $K$ that lies in the union of the horizontal plane and a finite union of vertical planes so that the unit tangent vector to $K$ is never vertical is *almost-horizontal*. The *writhe* of a generic almost-horizontal knot embedding is the number of positive crossings minus the number of negative crossings of its orthogonal projection onto the horizontal plane.<fnref n="4" /> An almost-horizontal embedding $K$ has a natural parallel $K_{\parallel}$ (up to isotopy) obtained from $K$ by (slightly) pushing it down.<fnref n="5" /> For any almost-horizontal knot embedding $K$, the degree of $G_K$ extends to a constant function of $S^2$. More precisely, we have the following lemma.

**Lemma 1.6.** For any almost-horizontal knot embedding $K$, the degree of $G_K$ at any regular value of $G_K$ is the writhe of $K$, and we have

$$I_{\theta}(K) = \int_{\tilde{C}(K; \to)} G_{K}^{*}(\omega_{S^{2}}) = lk(K, K_{\parallel}).$$

**PROOF:** As in the previous examples, for such a knot embedding $K$, the degree of $G_K$ extends to a constant function on each hemisphere of $S^2$. It maps the regular values of these hemispheres to the writhe of $K$. In the complement of $K$, the parallel $K_{\parallel}$ is isotopic to the parallel $K_{\parallel, \ell}$ on the left-hand side of $K$. The formulas of Subsection 1.2.1 show that $lk(K, K_{\parallel, \ell})$ is the writhe of $K$. $\square$

A *knot invariant* is a function of embeddings that takes the same value on isotopic knots. Unlike the Gauss linking number, the integral $I_{\theta}()$ is *not invariant under isotopy* since it takes distinct values on the isotopic knot embeddings $K_{-1}$ and $K_1$.

In every isotopy class of embeddings of $S^1$ into $\mathbb{R}^3$, we can construct an embedding $K$ such that the degree of the direction map $G_K$ extends as the constant map of $S^2$ with value 0 (or any arbitrary integer) as follows:

<footnote n="4">⁴The genericity of $K$ implies that the orthogonal projection onto the horizontal plane of $K$ is an immersion whose only multiple points are transverse double points.</footnote>

<footnote n="5">⁵A *parallel* of a knot embedding $K$ in a 3-manifold $M$ (such as $\mathbb{R}^3$) is a knot embedding $K_{\parallel}: S^1 \hookrightarrow M$ such that there exists an embedding $f$ from $[0, 1] \times S^1$ into $M$ that restricts to $\{0\} \times S^1$ as $K$ and to $\{1\} \times S^1$ as $K_{\parallel}$.</footnote>
add kinks such as $\rightsquigarrow$ or $\rightsquigarrow$ to a generic horizontal projection, and take a corresponding almost-horizontal embedding.

Since $I_\theta$ varies continuously under an isotopy, it maps any isotopy class of embeddings of $S^1$ into $\mathbb{R}^3$ onto $\mathbb{R}$. In particular, there are embeddings $K$ for which $I_\theta(K)$ is not an integer. For such an embedding, the degree of the direction map $G_K$ cannot be extended to a constant map on $S^2$.

### 1.2.5 A first knot invariant which counts graph configurations

Let us now try to count configurations $c$ of the following graph:

<figure><img src="image_17.png" /></figure>

A configuration $c$ of that graph is an injection from the set $\{b_3, a_2, a_3, b_2\}$ of vertices of $\mathcal{X}$ to $\mathbb{R}^3$ such that the images $c(b_3)$, $c(a_2)$, $c(a_3)$, and $c(b_2)$ of the vertices are on the knot $K$, and we successively meet $c(b_3) = K(z)$, $c(a_2) = K(z \exp(2i\pi\alpha_2))$, $c(a_3) = K(z \exp(2i\pi\alpha_3))$, and $c(b_2) = K(z \exp(2i\pi\beta_2))$ along $K$ following the orientation of $K$. The dashed circle shows the cyclic order of the four vertices. The associated configuration space $\tilde{C}(K; \mathcal{X})$ is

$$ \left\{ \left( K(z), K(z \exp(2i\pi\alpha_2)), K(z \exp(2i\pi\alpha_3)), K(z \exp(2i\pi\beta_2)) \right) : z \in S^1, (\alpha_2, \alpha_3, \beta_2) \in ]0, 1]^3, \alpha_2 < \alpha_3 < \beta_2 \right\}. $$

For $i \in \{2, 3\}$, set $e_i = (a_i, b_i)$, and let $G_{e_i}(c) = \frac{c(b_i) - c(a_i)}{\|c(b_i) - c(a_i)\|}$ denote the direction in $S^2$ of the image under the configuration $c$ of the edge $e_i$. The open configuration space $\tilde{C}(K; \mathcal{X})$ has the natural compactification

$$ C(K; \mathcal{X}) = S^1 \times \{(\alpha_2, \alpha_3, \beta_2) \in [0, 1]^3 : \alpha_2 \le \alpha_3 \le \beta_2\}. $$

The maps $G_{e_2}$ and $G_{e_3}$ smoothly extend to $C(K; \mathcal{X})$ as before, and

$$ G_{\mathcal{X}} = (G_{e_2}, G_{e_3}) : C(K; \mathcal{X}) \to (S^2)^2 $$

is a smooth map between two compact 4-manifolds.

The codimension-one faces $C(K; \mathcal{X})$ are the four faces $(\alpha_2 = 0)$, $(\alpha_2 = \alpha_3)$, $(\alpha_3 = \beta_2)$, and $(\beta_2 = 1)$, on which $c$ maps (at least) two consecutive vertices to the same point on $K$.<fnref n="6" /> When $G_{\mathcal{X}}$ is locally an embedding near

<footnote n="6">⁶In this introduction, codimension-one faces are closed. Later, they will be open parts of the boundaries.</footnote>
such a face, the degree of $G_{\mathcal{K}}$ changes by $\pm 1$ when we cross the image of that face.<fnref n="7" /> Thus, it suffices to determine the images of the interiors of these codimension-one faces and the local degree at one regular value to determine the degree of $G_{\mathcal{K}}$, as a map from $(S^2)^2 \setminus G_{\mathcal{K}}(\partial C(K; \mathcal{K}))$ to $\mathbb{Z}$. We associate the following figure to the face $(\beta_2 = 1)$ of $\tilde{C}(K; \mathcal{K})$.

<figure><img src="image_18.png" /></figure>

Let us now try to count configurations $c$ of the following tripod $\mathcal{K}$.

<figure><img src="image_19.png" /></figure>

A configuration $c$ of this tripod $\mathcal{K}$ is an injection from the set $\{w, v_1, v_2, v_3\}$ of its vertices into $\mathbb{R}^3$, where the images $c(v_1)$, $c(v_2)$, and $c(v_3)$ of the vertices $v_1$, $v_2$, and $v_3$ are on the knot $K$, and we successively meet $c(v_1)$, $c(v_2)$, and $c(v_3)$ along $K$ following the orientation of $K$.

<figure><img src="image_20.png" /><figcaption>Figure 1.7: A configuration of the tripod on $K$</figcaption></figure>

Such a configuration $c$ maps $w$ to $c(w) \in \mathbb{R}^3$, $v_1$ to $c(v_1) = K(z)$ for some $z \in S^1$, $v_2$ to $c(v_2) = K(z \exp(2i\pi t_2))$, and $v_3$ to $c(v_3) = K(z \exp(2i\pi t_3))$. The set of these configurations is the configuration space $\tilde{C}(K; \mathcal{K})$. It is an open 6-manifold parametrized by an open subspace of $\mathbb{R}^3 \times S^1 \times \{(t_2, t_3) \in ]0, 1]^2 : t_2 < t_3\}$. For $i \in \underline{3} = \{1, 2, 3\}$, set $e_i = (v_i, w)$, and let $G_{e_i}(c) = \frac{c(w)-c(v_i)}{\|c(w)-c(v_i)\|}$ denote the direction of the image under $c$ of the edge $e_i$ in $S^2$. These edge directions together provide a map

$$ \check{G}_{\mathcal{K}}: \begin{array}{ccc} \check{C}(K; \mathcal{K}) & \rightarrow & (S^2)^3 \\ c & \mapsto & (G_{e_1}(c), G_{e_2}(c), G_{e_3}(c)). \end{array} $$

from our open 6-manifold $\check{C}(K; \mathcal{K})$ to the 6-manifold $(S^2)^3$.

<footnote n="7">⁷See Lemma 2.3 for a precise statement.</footnote>
For a regular value $(X_1, X_2, X_3)$ of $\tilde{G}_{\mathcal{S}}$ whose preimage is finite, we can again count the configurations of the tripod such that the direction of the edge $e_i$ is $X_i$, as the degree of $\tilde{G}_{\mathcal{S}}$ at $(X_1, X_2, X_3)$.

In Chapter 8, we construct a compactification $C(K; \mathcal{S})$ of $\tilde{C}(K; \mathcal{S})$ (and of many similar configuration spaces), using blow-up techniques, as William Fulton, Robert MacPherson [FM94], Scott Axelrod, Isadore Singer [AS92], and Maxim Kontsevich [Kon94] did. This compactification $C(K; \mathcal{S})$ is a smooth compact 6-manifold with boundary and ridges. Its interior is $\tilde{C}(K; \mathcal{S})$. The map $\tilde{G}_{\mathcal{S}}$ extends to a smooth map $G_{\mathcal{S}}$ over $C(K; \mathcal{S})$. Regular values of $G_{\mathcal{S}}$ are regular values of $\tilde{G}_{\mathcal{S}}$ in the complement of $G_{\mathcal{S}}(\partial C(K; \mathcal{S})) = C(K; \mathcal{S}) \setminus \tilde{C}(K; \mathcal{S})$. They form an open dense subset $\mathcal{O}$ of $(S^2)^3$, for which the differential degree of $\tilde{G}_{\mathcal{S}}$ makes sense. As mentioned above and proved in Lemma 2.3, this local integral degree extends to a continuous map on $(S^2)^3 \setminus G_{\mathcal{S}}(\partial C(K; \mathcal{S}))$. It restricts to a constant map on every connected component of $(S^2)^3 \setminus G_{\mathcal{S}}(\partial C(K; \mathcal{S}))$. In Lemma 7.12, we explicitly compute this local degree when $K$ is the round circle $\mathcal{O}$ in a plane, as an example. Our computation shows that this degree cannot be extended to a constant map in this case.

Let $p_i: (S^2)^3 \to S^2$ denote the projection to the $i^{th}$ factor. Define the algebraic volume

$$I(K; \mathcal{S}) = \int_{(S^2)^3} \deg(G_{\mathcal{S}}) \wedge_{i=1}^3 p_i^*(\omega_{S^2}) = \int_{\tilde{C}(K; \mathcal{S})} G_{\mathcal{S}}^* (\wedge_{i=1}^3 p_i^*(\omega_{S^2}))$$

of the image of $C(K; \mathcal{S})$ under $G_{\mathcal{S}}$. This algebraic volume $I(K; \mathcal{S})$ has no reason to be a knot isotopy invariant, and it is not.

To compute the map

$$\deg(G_{\mathcal{S}}): (S^2)^3 \setminus G_{\mathcal{S}}(\partial C(K; \mathcal{S})) \to \mathbb{Z},$$

we look at $G_{\mathcal{S}}(\partial C(K; \mathcal{S}))$. Let us describe the compactification $C(K; \mathcal{S})$ on loci where $c(w)$ approaches $c(v_1)$ and $c(v_1)$ is far from $c(v_2)$ and $c(v_3)$. With our coordinates, this amounts to assuming $c(w) = c(v_1) + \eta x$ for some $\eta \in ]0, \varepsilon]$ for a small $\varepsilon > 0$ and some $x \in S^2$, and $(t_2, t_3) \in [\alpha, 1-\alpha]^2$ for some $\alpha \in ]0, 1/2[$. This part of $\tilde{C}(K; \mathcal{S})$ is diffeomorphic to $]0, \varepsilon] \times S^2 \times S^1 \times \{(t_2, t_3) \in [\alpha, 1-\alpha]^2 : t_2 < t_3\}$. Its closure in the compactification $C(K; \mathcal{S})$ is naturally diffeomorphic to $]0, \varepsilon] \times S^2 \times S^1 \times \{(t_2, t_3) \in [\alpha, 1-\alpha]^2 : t_2 \le t_3\}$. (We close $]0, \varepsilon[$ at 0—we also relax the inequality $t_2 < t_3$ but this is not important for us now.) In the compactification $C(K; \mathcal{S})$, the image $c(w)$ may coincide with $c(v_1)$ (when $\eta = 0$). The direction from $c(v_1)$ to $c(w)$ is still defined in this case. It is contained in the $S^2$ factor. In particular, $G_{\mathcal{S}}$ extends to this part of $C(K; \mathcal{S})$. The compactification creates the local
boundary {$0$} $\times$ ($-S^2$) $\times$ $S^1$ $\times$ {$(t_2, t_3) \in ]\alpha, 1 - \alpha]^2 : t_2 \le t_3$}. This local boundary is a 5-dimensional manifold. Its image under $G_{\mathcal{S}}$ is the product by $S^2$ of the image in $(S^2)^2$ of the configuration space associated to the following graph by the natural Gauss map "direction of the edges".

<figure><img src="image_21.png" /></figure>

Again, the dashed circle represents the cyclic order of $c(v_1)$, $c(v_2)$, and $c(v_3)$ along $K$. The image of this local boundary creates a "wall" in $(S^2)^3$ across which the local degree changes by $\pm 1$. (See Lemma 2.3 for a precise statement.)

We recognize the picture associated to the face $(\beta_2 = 1)$ denoted by $F_{\beta_2=1}$ of $C(K; \mathcal{S})$. We observe that the image of the corresponding face under $G_{\mathcal{S}}$ coincides with the image of the face $S^2 \times F_{\beta_2=1}$ of $S^2 \times C(K; \mathcal{S})$ under

$$G'_{\mathcal{S}} = 1_{S^2} \times G_{\mathcal{S}} : S^2 \times C(K; \mathcal{S}) \to (S^2)^3.$$

So the combination $(\deg(1_{S^2} \times G_{\mathcal{S}}) - \deg(G'_{\mathcal{S}}))$ does not vary across the images of the corresponding faces (or at least not because of them). (There are some sign and orientation issues to check here, but we will carefully treat them in a broader generality in Section 7.1 and Lemma 9.14. Let the reader trust me that the signs are correct here.) We glued the images of $G'_{\mathcal{S}}$ and of $G_{\mathcal{S}}$ along $G'_{\mathcal{S}}(S^2 \times F_{\beta_2=1})$ to make the union of these images behave as the image of a manifold without boundary, locally. Unfortunately, the three other faces of $S^2 \times C(K; \mathcal{S})$ created other walls in $(S^2)^3$ associated to the following figures:

<figure><img src="image_22.png" /></figure>

To cancel these walls with the same type of faces of $C(K; \mathcal{S})$ as before, we use Gauss maps associated to the following diagrams:

<figure><img src="image_23.png" /></figure>

Let us describe these Gauss maps more precisely. For any subset $I$ of $\mathcal{S}$, let $\iota_I$ denote the diffeomorphism of $(S^2)^3$ which maps $(X_1, X_2, X_3)$ to $(\varepsilon_1 X_1, \varepsilon_2 X_2, \varepsilon_3 X_3)$, where $\varepsilon_i = -1$ when $i \in I$, and $\varepsilon_i = 1$ when $i \notin I$. For two elements $i, j$ of $\mathcal{S}$, simply write $\iota_i = \iota_{\{i\}}$ and $\iota_{ij} = \iota_{\{i,j\}}$. For a permutation $\sigma$ of $\mathcal{S}$, let $\sigma_*$ denote the diffeomorphism of $(S^2)^3$ that maps $(Y_1, Y_2, Y_3)$ to $(Y_{\sigma^{-1}(1)}, Y_{\sigma^{-1}(2)}, Y_{\sigma^{-1}(3)})$. (We have $\sigma_*((X_{\sigma(1)}, X_{\sigma(2)}, X_{\sigma(3)})) = (X_1, X_2, X_3)$.)
The Gauss maps associated to the above diagrams are $\iota_2 \circ (23)_* \circ G_{\mathcal{S}}$, $\iota_{23} \circ G_{\mathcal{S}}$, and $\iota_3 \circ (23)_* \circ G_{\mathcal{S}}$, respectively. The combination

$$ \deg(1_{S^2} \times G_{\mathcal{S}}) - \deg(G_{\mathcal{S}}) - \deg(\iota_{23} \circ G_{\mathcal{S}}) \\ + \deg(\iota_2 \circ (23)_* \circ G_{\mathcal{S}}) + \deg(\iota_3 \circ (23)_* \circ G_{\mathcal{S}}) $$

does not vary across the boundary of the image of $1_{S^2} \times G_{\mathcal{S}}$ as Figure 1.8 suggests.

<figure><img src="image_24.png" /><figcaption>Figure 1.8: Variation of the image of $1_{S^2} \times G$ across the boundary in $(S^2)^3$ (pictured as 2-dimensional)</figcaption></figure>

Unfortunately, this process introduces other walls, and we have not yet cancelled the walls due to the faces $c(v_2) = c(w)$ and $c(v_3) = c(w)$. However, we have the following proposition, which is a corollary of Theorem 7.32, as it will be seen right after Theorem 7.32.

**Proposition 1.7.** The map $\tilde{w}_2$

$$ \frac{1}{24} \sum_{\sigma \in \mathfrak{S}_3} \deg \left( \sigma_*(1_{S^2} \times G_{\mathcal{S}}) \right) \\ - \frac{1}{48} \sum_{I \subseteq \underline{3}} (-1)^{|I|} \left( \deg(\iota_I \circ G_{\mathcal{S}}) + \deg(\iota_I \circ (23)_* \circ G_{\mathcal{S}}) \right), $$

which is well-defined on an open dense subset of $(S^2)^3$, extends as a constant function of $(S^2)^3$ whose value $w_2(K)$ is in $\frac{1}{48}\mathbb{Z}$.

**SKETCH OF PROOF:** Let us show that the boundary

$$ \sigma_*(S^2 \times G_{\mathcal{S}}(\partial C(K; \mathcal{S}))) $$
can be glued to the images of the faces $c(v_i) = c(w)$ of $C(K; \mathcal{X})$ under the $\iota_I \circ G_{\mathcal{X}}$ or the $\iota_I \circ (23)_* \circ G_{\mathcal{X}}$, up to sign. For every permutation $\sigma$ of $\underline{3}$, the boundary $\sigma_* (S^2 \times G_{\mathcal{X}}(\partial C(K; \mathcal{X})))$ consists of the four faces

<figure><img src="image_25.png" /></figure>

where the first one and the third one come with a coefficient $-\frac{1}{24}$, and the second one and the fourth one come with a coefficient $\frac{1}{24}$. The images of the (open) faces $(c(v_1) = c(w))$, $(c(v_2) = c(w))$, and $(c(v_3) = c(w))$ under $G_{\mathcal{X}}$ are

<figure><img src="image_26.png" /></figure>

The images of the (open) faces $(c(v_1) = c(w))$, $(c(v_2) = c(w))$, and $(c(v_3) = c(w))$ under $(23)_* \circ G_{\mathcal{X}}$ are

<figure><img src="image_27.png" /></figure>

In order to obtain the images under the compositions of these maps by some $\iota_I$, we reverse the edge $i$ when $i \in I$. Each of these faces appears twice (once for each orientation of the collapsed edge), with the same sign (since the antipodal map of $S^2$ reverses the orientation) with a coefficient $(-1)^{n(F)} \frac{1}{48}$, where $n(F)$ is the number of edges towards the bivalent vertex.

We leave the general discussion of signs to the reader, and we do not discuss all the faces of $C(K; \mathcal{X})$ in this sketch. Let us just mention that the image of the faces $c(v_1) = c(v_2)$ is contained in the codimension-two subspace of $(S^2)^3$, for which at least two $S^2$-coordinates are equal or opposite. So these faces do not create walls and may be forgotten. $\square$

The $\sigma_*$ preserve the volume of $(S^2)^3$, and the $\iota_I$ multiply the volume by $(-1)^{|I|}$. Therefore, the combination $w_2(K)$ in Proposition 1.7 may also be written as

$$w_2(K) = \frac{1}{4} \int_{C(K; \mathcal{X})} G_{\mathcal{X}}^* (\wedge_{i=1}^2 p_i^*(\omega_{S^2})) - \frac{1}{3} \int_{C(K; \mathcal{X})} G_{\mathcal{X}}^* (\wedge_{i=1}^3 p_i^*(\omega_{S^2})).$$

Since $w_2(.)$ is valued in $\frac{1}{48}\mathbb{Z}$, and since $w_2(K)$ varies continuously under an isotopy of $K$, $w_2(.)$ is an isotopy invariant. Enore Guadagnini, Maurizio Martellini, and Mihail Mintchev [GMM90], and Dror Bar-Natan [BN95b] independently studied the invariant $w_2(.)$ under this integral form in 1990.
To prove its isotopy invariance, one can alternatively use Stokes' theorem to evaluate the variations of the integrals under a knot isotopy $(t, z) \mapsto K_t(z)$. In [BT94], Raoul Bott and Clifford Taubes used compactifications of the one-parameter configuration spaces $\cup_{t \in [0,1]} C(K_t; \Gamma)$, like those discussed above, to check invariance.

The above formulation of Proposition 1.7 presents the invariant $w_2(K)$ as a discrete count of configurations of $\mathcal{K}$ and $\mathcal{K}$, as Dylan Thurston [Thu99] and Sylvain Poirier [Poi02] first did independently.<fnref n="8" /> For a generic triple $(a, b, c)$ of $(S^2)^3$, we count the configurations of $\mathcal{K}$ for which the edge directions are a pair of noncolinear vectors in $\{a, b, c, -a, -b, -c\}$, and the configurations of $\mathcal{K}$ for which the edge directions are a triple of pairwise noncolinear vectors in $\{a, b, c, -a, -b, -c\}$, with some coefficients and some signs determined by the corresponding above local degrees.

Any knot of $\mathbb{R}^3$ is obtained from the trivial knot by (isotopies and) a finite number of crossing changes $\mathcal{K} \to \mathcal{K}$. So, in order to determine a real-valued knot invariant $w$, it suffices to know its value on the trivial knot and its variation $w(\mathcal{K}) - w(\mathcal{K})$, denoted by $w(\mathcal{K})$, under crossing change. (Here, $\mathcal{K}$ and $\mathcal{K}$ represent knot diagrams that coincide outside a disk that they intersect as in the figure, and $\mathcal{K}$ represents a diagram that coincides with the former diagrams outside this disk.)

The variation $w(\mathcal{K}) = w(\mathcal{K}) - w(\mathcal{K})$ can be thought of as a discrete derivative of the invariant. The variation of this variation

$$ \begin{aligned} w(\mathcal{K} \mathcal{K}) &= w(\mathcal{K} \mathcal{K}) & -w(\mathcal{K} \mathcal{K}) \\ &= w(\mathcal{K} \mathcal{K}) - w(\mathcal{K} \mathcal{K}) & -w(\mathcal{K} \mathcal{K}) + w(\mathcal{K} \mathcal{K}) \end{aligned} $$

under a disjoint crossing change is thought of as a discrete second derivative of this invariant. In $\mathbb{R}^3$, a real-valued knot invariant $w$ is actually determined by its value on the trivial knot and the discrete second derivative. (If the second derivative is zero, then the variation under a crossing change is independent of the knot, and it is the same as $(w(\mathcal{K}) - w(\mathcal{K})) = 0$.) In [GMM90], Enore Guadagnini, Maurizio Martellini, and Mihail Mintchev computed $w_2(O) = -\frac{1}{24}$. This is reproved in Examples 7.11 and Lemma 7.12. In [BN95b, §6.3], Dror Bar-Natan computed the above "discrete second derivative" for the invariant $w_2$, and he found:

$$ w_2(\mathcal{K} \mathcal{K}) = 0 \quad \text{and} \quad w_2(\mathcal{K} \mathcal{K}) = 1, $$

where the dashed lines indicate the connections inside $K$ of the crossing strands. This allowed him to identify $w_2$ with $a_2 - \frac{1}{24}$, where $a_2(K) =$

<footnote n="8">⁸Michael Polyak and Oleg Viro obtained a similar result [PV01, Theorem 3.A, Section 3.5] in the setting of long knots of $\mathbb{R}^3$, with fewer involved gluings, and with induced combinatorial formulae in terms of knot diagrams.</footnote>
$\frac{1}{2}\Delta''(K)(1)$ is half the second derivative of the Alexander polynomial of $K$ at one, since $a_2$ has the same discrete second derivative as $w_2$ and $a_2(O) = 0$.

### 1.2.6 On other similar invariants of knots in $\mathbb{R}^3$

We can associate similar integrals over configuration space to every uni-trivalent graph $\Gamma$ whose univalent vertices are ordered cyclically, as in the following figure:

<figure><img src="image_28.png" /></figure>

We can also exhibit other similar combinations that provide isotopy invariants of knots in $\mathbb{R}^3$, as several authors, including Maxim Kontsevich [Kon94], Daniel Altschüler and Laurent Freidel [AF97], Dylan Thurston [Thu99], did. These invariants are all finite type invariants with respect to the following definition. An invariant valued in an abelian group is of degree less than $n$ if all its discrete derivatives of order $n$, which generalize the previously studied discrete second derivative, vanish. A knot invariant is a *Vassiliev invariant* or a *finite type invariant* if it is of degree less than some integer $n$. (A precise definition is given in Section 6.1.) In [Kon93, BN95a], Dror Bar-Natan and Maxim Kontsevich proved the “fundamental theorem of Vassiliev invariants”, which determines the space of real-valued finite type invariants as the dual of an algebra $\mathcal{A}$ generated by uni-trivalent graphs. Daniel Altschüler and Laurent Freidel [AF97]—and Dylan Thurston [Thu99] independently—constructed a “universal Vassiliev invariant” $\mathcal{Z}$ of knots in $\mathbb{R}^3$, valued in $\mathcal{A}$ such that, for any knot $K$ of $\mathbb{R}^3$, $\mathcal{Z}(K)$ is a combination of classes of uni-trivalent graphs whose coefficients are integrals over corresponding configuration spaces.<fnref n="9" /> Any real-valued finite type invariant may be expressed as $\psi \circ \mathcal{Z}$ for some linear form $\psi: \mathcal{A} \to \mathbb{R}$. Therefore, every real-valued finite type knot invariant is a combination of integrals over configuration spaces associated to uni-trivalent graphs. It is still unknown whether finite type knot invariants distinguish all knots of $\mathbb{R}^3$, but many known polynomial invariants of knots of $\mathbb{R}^3$, such as the Alexander polynomial, the Jones polynomial, and the HOMFLYPT polynomial, factor through $\mathcal{Z}$.

<footnote n="9">⁹The original proof of the fundamental theorem of Vassiliev invariants relies on the construction of another—possibly equal—universal Vassiliev invariant called the *Kontsevich integral*.</footnote>
### 1.2.7 On similar invariants of knots in other 3-manifolds

Recall that $D_1 \times [0, 1]$ denotes the standard cylinder in $\mathbb{R}^3 = \mathbb{C} \times \mathbb{R}$, and that a rational homology cylinder $\mathcal{C}$ is a compact oriented 3-manifold with the same boundary and the same rational homology as $D_1 \times [0, 1]$ (i.e., the rational homology of a point).<fnref n="10" />

In this book, following ideas of Maxim Kontsevich [Kon94], Greg Kuperberg, and Dylan Thurston [KT99], we generalize $\mathcal{Z}$ to links in more general 3-manifolds denoted by $\tilde{R} = \tilde{R}(\mathcal{C})$. These manifolds are constructed from $\mathbb{R}^3$ by replacing $D_1 \times [0, 1]$ by a rational homology cylinder $\mathcal{C}$. Such a more general 3-manifold $\tilde{R}$ is called a rational homology $\mathbb{R}^3$. It looks like $\mathbb{R}^3$ near $\infty$. The linking number can be defined as follows for a two-component link embedding $J \cup K: S^1 \cup S^1 \to \tilde{R}$.

Two submanifolds $A$ and $B$ in a manifold $M$ are transverse if at each intersection point $x \in A \cap B$, we have $T_x M = T_x A + T_x B$. If two transverse oriented submanifolds $A$ and $B$ in an oriented manifold $M$ are of complementary dimensions (i.e., if the sum of their dimensions is the dimension of $M$), then the sign of an intersection point is +1 if $T_x M = T_x A \oplus T_x B$ as oriented vector spaces. Otherwise, the sign is -1. If $A$ and $B$ are compact, and if $A$ and $B$ are of complementary dimensions in $M$, then their algebraic intersection is the sum of the signs of the intersection points, it is denoted by $\langle A, B \rangle_M$.

When $K$ bounds a compact oriented embedded surface $\Sigma_K$ in $\tilde{R}$ transverse to $J$, the linking number of $J$ and $K$ is the algebraic intersection number $\langle J, \Sigma_K \rangle_{\tilde{R}}$ of $J$ and $\Sigma_K$ in $\tilde{R}$. In general, there is an oriented surface $\Sigma_{nK}$ immersed in $\tilde{R}$ whose (oriented!) boundary is a positive multiple $nK$ of $K$, and $lk(J, K) = \frac{1}{n}\langle J, \Sigma_{nK} \rangle_{\tilde{R}}$.

For two-component links in $\mathbb{R}^3$, this definition coincides with Definition 1.1 of the Gauss linking number. See Proposition 2.9.

In the more general setting of a rational homology $\mathbb{R}^3$, instead of counting uni-trivalent graphs whose edge directions belong to a finite set of directions, we use the notion of propagator defined in Chapter 3. A propagator is a rational combination of oriented compact 4-manifolds in a suitable compactification $C_2(R)$ (defined in Section 3.2) of the configuration space

$$ \tilde{C}_2(R) = \{(x, y) \in \tilde{R}^2 : x \neq y\}, $$

<footnote n="10">¹⁰This homological condition can be rephrased as “The (compact oriented) 3-manifold $\mathcal{C}$ is connected, and every knot embedding $K$ in $\mathcal{C}$ bounds a rational chain in $\mathcal{C}$.”. Let $N(K)$ be a compact tubular neighborhood of $K$ in $\mathcal{C}$, and let $\tilde{N}(K)$ denote its interior. The latter condition is equivalent to the existence of a compact oriented surface in $\mathcal{C} \setminus \tilde{N}(K)$ whose (oriented!) boundary is a disjoint union of curves in $\partial N(K)$ that does not bound a compact oriented surface in $N(K)$.</footnote>
which shares many properties with our *model propagator*

$$p_{S^2}^{-1}(X) = \overline{\{(x, x + tX) : x \in \mathbb{R}^3, t \in ]0, +\infty[\}}$$

for $X \in S^2$ (and $\tilde{R} = \mathbb{R}^3$).

With these model propagators $p_{S^2}^{-1}(X)$, the direction of a configured edge $(c(x), c(y))$ is $X$ if and only if $(c(x), c(y))$ belongs to the propagator $p_{S^2}^{-1}(X)$. More general propagators produce similar codimension-two constraints on configurations. They allow us to count uni-trivalent graphs whose configured edges belong to a finite set of propagators, with signs, as above.

In general, the boundary of a propagator of $C_2(R)$ is in the boundary of $C_2(R)$, and, for any two-component link embedding $J \sqcup K: S^1 \sqcup S^1 \to \tilde{R}$, the algebraic intersection of $J \times K \subset \tilde{C}_2(R)$ with a propagator in $C_2(R)$ is the linking number of $J$ and $K$. (Note that the linking number of two knots in $\mathbb{R}^3$ is indeed this algebraic intersection with a model propagator.)

### 1.2.8 Morse propagators

Let us show examples of propagators in a general rational homology $\mathbb{R}^3$.

View a rational homology $\mathbb{R}^3$ as the union of two genus $g$ handlebodies as in Figure 1.9, where the two pieces are glued to each other by an a priori nontrivial diffeomorphism of $\partial H_a$.<fnref n="11" />

<figure><img src="image_29.png" /><figcaption>Figure 1.9: $H_a$ and $H_b$</figcaption></figure>

<footnote n="11">¹¹Unlike the handlebodies in the rest of this book, which are as in Section 1.1, the handlebodies of this section are not compact.</footnote>
The handlebody $H_a$ has $g$ arbitrarily oriented meridian disks $D(\alpha_i)$ centered at $a_i$, for $i$ in $\underline{g} = \{1, 2, \dots, g\}$. The handlebody $H_b$ has $g$ arbitrarily oriented meridian disks $D(\beta_j)$ centered at $b_j$, for $j \in \underline{g}$. The topology of $\check{R}$ is determined by the curves $\alpha_i = \partial D(\alpha_i)$ and $\beta_j = \partial D(\beta_j)$ in the surface $\partial H_a$. The data $(\partial H_a, (\alpha_i)_{i \in \underline{g}}, (\beta_j)_{j \in \underline{g}})$ is called a Heegaard diagram. From such data, we construct

* a Morse function $f: \check{R} \to \mathbb{R}$ such that $\partial H_a = f^{-1}(1/2)$ and the critical points of $f$ (the points at which the derivative of $f$ vanishes) are the $a_i$, which have index one and are in $f^{-1}(1/3)$, and the $b_j$, which have index two and are in $f^{-1}(2/3)$,

* a gradient vector field $\nabla: \check{R} \to T\check{R}$ associated to $f$ and some metric $\mathfrak{g}$ such that

$$T_x f(y \in T_x \check{R}) = \langle \nabla(x), y \rangle_{\mathfrak{g}},$$

and

* an associated gradient flow $\phi: \mathbb{R} \times \check{R} \to \check{R}$ such that $\phi(0, \cdot)$ is the Identity map and $\frac{\partial}{\partial t}\phi(t, x)_{(u,x)} = \nabla(\phi(u, x)).$

Let $\mathcal{S}$ be the set of critical points of $f$. The flow lines of the flow $\phi$ are the $\phi(\mathbb{R} \times \{y\})$, for the $y$ in the complement $\check{R} \setminus \mathcal{S}$ in $\check{R}$ of $\mathcal{S}$. For the standard height function $f_0$ and the standard metric of $\mathbb{R}^3$, the associated flow is

$$((\phi_t = \phi(t, \cdot)): x \mapsto x + t\vec{N}),$$

where $\vec{N} = (0, 0, 1)$. Our rational homology $\mathbb{R}^3$ (equipped with the metric $\mathfrak{g}$) is assumed to coincide with $\mathbb{R}^3$ (equipped with its standard metric) outside $D_1 \times [0, 1]$, and our Morse function (can be and) is assumed to coincide with $f_0$ outside $D_1 \times [0, 1]$.

As a set, the complement $\check{R} \setminus \mathcal{S}$ of the set of critical points is the disjoint union of the flow lines diffeomorphic to $\mathbb{R}$, which behave as follows. There are two flow lines $\mathcal{L}_+(a_i)$ and $\mathcal{L}_-(a_i)$ starting as vertical lines and ending at $a_i$ that approach $a_i$ at $+\infty$, as in Figure 1.10. (They start as vertical lines $\{\cdot\} \times ]-\infty, 0[.$) The closure in $\check{R}$ of their union is a line $\mathcal{L}(a_i)$, called the *descending manifold* of $a_i$. It is oriented so that its algebraic intersection with $D(\alpha_i)$ is 1. (This is not consistent with the bottom-top orientation of one of the flow lines. If the flow line with the orientation of the positive normal to $D(\alpha_i)$ is $\mathcal{L}_+(a_i)$, then $\mathcal{L}(a_i) = \overline{\mathcal{L}_+(a_i)} \cup (-\mathcal{L}_-(a_i))$. ) There are two flow lines $\mathcal{L}_+(b_j)$ and $\mathcal{L}_-(b_j)$ that approach $b_j$ at $-\infty$. The closure of

<footnote n="12">¹²The definition of this flow is justified in [Spi79, Chapter5], for example.</footnote>
<figure><img src="image_30.png" /><figcaption>Figure 1.10: $\mathcal{L}_+(a_i)$, $\mathcal{L}_-(a_i)$, $\mathcal{L}_+(b_j)$, $\mathcal{L}_-(b_j)$</figcaption></figure>

their union is a line $\mathcal{L}(b_j)$, which is called the *ascending manifold* of $b_j$. It is oriented so that its algebraic intersection with $D(\beta_j)$ is 1.

The closure of the union of the flow lines that approach $a_i$ at $-\infty$ is called the *ascending manifold* of $a_i$. It is denoted by $\mathcal{A}_i$. Its intersection with $H_a$ is $D(\alpha_i)$, and it is oriented like $D(\alpha_i)$. The closure of the union of the flow lines that approach $b_j$ at $+\infty$ is called the *descending manifold* of $b_j$. It is denoted by $\mathcal{B}_j$. Its intersection with $H_b$ is $D(\beta_j)$, and it is oriented like $D(\beta_j)$. The ascending manifold $\mathcal{A}_i$ is an immersion of Figure 1.11, which restricts to its interior as an embedding, where the flow lines $\gamma(c_k)$ are flow lines through crossings $c_k$ of $\alpha_i \cap \beta_j$, which approach $a_i$ near $-\infty$, and $b_j$ near $+\infty$. A figure for $\mathcal{B}_j$ is obtained by reversing the arrows and changing $a_i$ to $b_j$.

Except for the flow lines of the descending manifolds of the $a_i$ and the flow lines of the ascending manifolds of the $b_j$, each flow line intersects $\partial H_a$ once, transversally, with a positive sign.

<figure><img src="image_31.png" /><figcaption>Figure 1.11: The interior of $\mathcal{A}_i$.</figcaption></figure>

The flow lines not in the above descending or ascending manifolds begin as vertical half-lines $x \times ]-\infty, 0[$. They end as vertical half-lines $y \times ]1, \infty[$ for some $x, y$ in $\mathbb{R}^2$. Except for the critical points, every point has a neighborhood
diffeomorphic to a cube ]0, 1[^3 such that, with the induced identification, the flow maps $(t, x)$ to $x+t\vec{N}$, for any $(t, x)$ in $\mathbb{R}\times ]0, 1[^3$ such that $x+t\vec{N} \in ]0, 1[^3$.

In [Les15a, Theorem 4.2], Greg Kuperberg and I constructed a propagator from the gradient flow $(\phi_t = \phi(t, .))$ of a Morse function $f$ without minima and maxima as above, as follows. Let $P_\phi$ denote the closure in $C_2(\mathbb{R})$ of $\{(x, \phi_t(x)) : x \in \tilde{R} \setminus S, t \in ]0, +\infty[\}$). Note that when $(\phi_t : x \mapsto x + t\vec{N})$ is the flow associated to the standard height function $f_0$ of $\mathbb{R}^3$, $P_\phi = p_{S^2}^{-1}(\vec{N})$ is one of our model propagators. Let

$$[\mathcal{J}_{ji}]_{(j,i) \in \{1, \dots, g\}^2} = [\langle \alpha_i, \beta_j \rangle_{\partial H_a}]^{-1}$$

be the inverse matrix of the matrix of the algebraic intersection numbers $\langle \alpha_i, \beta_j \rangle_{\partial H_a}$. (This matrix is invertible because $\tilde{R}$ is a rational homology $\mathbb{R}^3$.)

Let $((\mathcal{B}_j \times \mathcal{A}_i) \cap C_2(\mathbb{R}))$ denote the closure of $((\mathcal{B}_j \times \mathcal{A}_i) \cap (\tilde{R}^2 \setminus \text{diagonal}))$ in $C_2(\mathbb{R})$, then

$$P(f, g) = P_{\phi} + \sum_{(i,j) \in \{1, \dots, g\}^2} \mathcal{J}_{ji}((\mathcal{B}_j \times \mathcal{A}_i) \cap C_2(\mathbb{R}))$$

is an example of a propagator.

Pick four small generic perturbations $P_1, P_2, P_3$, and $P_4$ of such a propagator (or of more general propagators as precisely defined in Section 3.3). The invariant $w_2$ can be extended to knots $K$ in $\tilde{R}$ as follows. Let $\Gamma$ be one of the graphs $\mathcal{K}$, $\mathcal{K}'$, or $\mathcal{K}''$. Orient the edges of $\Gamma$. Number them by the data of an injection $j_E$ from the set $E(\Gamma)$ of the (plain) edges of $\Gamma$ to $\mathbb{4}$. Thus, we can count the configurations of $\Gamma$ such that the configured oriented edge numbered by $i$ (viewed as the ordered pair of its ends) is in $P_i$, with signs precisely defined. Denote the average over the choices of such edge-orientations and numberings by $I_a(K, \Gamma, (P_1, P_2, P_3, P_4))$. (When averaging, we divide by the number $2^{|E(\Gamma)|}$ of edge orientations and by the number $\frac{4!}{(4-|E(\Gamma)|)!}$ of numberings.) Then

$$w_2(K) = I_a(K, \mathcal{K}, (P_1, P_2, P_3, P_4)) - I_a(K, \mathcal{K}', (P_1, P_2, P_3, P_4)) - 2I_a(K, \mathcal{K}''', (P_1, P_2, P_3, P_4))$$

does not depend on the chosen propagators. Furthermore, if $K$ is null-homologous, then $w_2(K)$ is again $\frac{1}{2}\Delta''(K)(1) - \frac{1}{24}$, in this more general setting, as we prove in Theorem 18.43.<fnref n="13" /> Note that when we compute $w_2(K \subset \mathbb{R}^3)$ with model propagators, $I_a(K, \mathcal{K}''', (P_1, P_2, P_3, P_4))$ vanishes because the double edge gives contradictory constraints.

<footnote n="13">¹³In [Let22], David Leturcq completely expresses the Alexander polynomial of long null-homologous knots in $\mathbb{Q}$-spheres in terms of similar counts of configurations.</footnote>
1.2.9 More about the contents of the book

The cited Altschüler-Freidel universal Vassiliev invariant of knots in $\mathbb{R}^3$ also extends to knots in a rational homology $\mathbb{R}^3$. This book describes this extended invariant $\mathcal{Z}$ using more general (and precisely defined) propagators. Kenji Fukaya proposed a way of counting configurations in a rational homology $\mathbb{R}^3$ in [Fuk96]. Other authors, including Tadayuki Watanabe [Wat18a] further studied his approach and made it rigorous. In this book, we can view the resulting way of counting as a particular way of counting with the above "Morse propagators" associated to Heegaard diagrams. These Morse propagators are examples of the general "propagating chains" described in Chapter 3. Counts of configurations as above also yield invariants of the ambient rational homology $\mathbb{R}^3$ and of its one-point compactification. This compactification is a *rational homology 3-sphere*, i.e., a connected oriented closed 3-manifold, where knots have a nontrivial multiple that bounds an immersed oriented compact surface.<fnref n="14" /> In [Mou12], Delphine Moussard developed a theory of finite type invariants for rational homology 3-spheres. Her theory involves surgery operations called *rational Lagrangian-preserving surgeries* instead of crossing changes. We describe these rational Lagrangian-preserving surgeries, which replace a piece of a manifold by another such, in Subsection 1.3.2 below. The above counts of configurations yield a universal finite type invariant $\mathcal{Z}$ of rational homology 3-sphere, with respect to Mousard's theory. The universality of $\mathcal{Z}$ follows from surgery formulae proved in [Les04b].

*Integer homology 3-spheres* are connected oriented closed 3-manifolds where knots bound an embedded oriented compact surface, as in the standard 3-sphere $S^3$. The invariant $\mathcal{Z}$ also restricts to a universal finite type invariant for integer homology 3-spheres, with respect to the Ohtsuki theory of finite type invariants [Oht96] and other equivalent theories described in [GGP01], as first shown by Greg Kuperberg and Dylan Thurston in [KT99].

Let us say a little more about the contents of this book, which is mostly self-contained. We describe the background material in the appendices. Unlike in this informal introduction, we will give details and precise statements and carefully check all the assertions. Our general invariant $Z$ is an infinite series of independent nontrivial invariants. To define it, we must equip any rational homology $\mathbb{R}^3$ with a parallelization $\tau: \tilde{R} \times \mathbb{R}^3 \to T\tilde{R}$. We describe the space of suitable parallelizations up to homotopy and the associated Pontjagin numbers in Chapter 5.

This book has four parts. In its present first part, we first state conven-

<footnote n="14">¹⁴A manifold is said to be *closed* if it is compact and connected, and if its boundary is empty.</footnote>
tions and known facts about 3-manifolds. Then we precisely define propagating chains associated with parallelizations and discuss the invariant $\Theta$ of parallelized rational homology 3-spheres. This invariant $\Theta$ is associated to the graph $\Theta$. It is the algebraic intersection of three transverse propagating chains. We explain how to get rid of the dependence of the parallelization of the rational homology 3-spheres with the help of relative Pontrjagin numbers to get an invariant of (unparallelized) rational homology 3-spheres in Section 4.3. This invariant is (six times) the Casson–Walker invariant of rational homology spheres. Andrew Casson and Kevin Walker originally defined it as a “count” of conjugacy classes of $SU(2)$-representations of the fundamental group of these manifolds [AM90, GM92, Mar88, Wal92].

Our general invariant $\mathcal{Z}^f$ is an invariant of parallelized links in rational homology 3-spheres.<fnref n="15" /> We define it in the second part of the book. It is valued in vector spaces generated by uni-trivalent graphs like $\Theta$, $\mathcal{K}$, and $\mathcal{X}$. We describe these spaces of diagrams and their rich structures, which help formulate the properties of the invariants, in Chapter 6. We present the general definition of $\mathcal{Z}^f$ for links in Chapter 7. We first give it in terms of integrals rather than in terms of discrete counts because the results are easier to write and prove in the world of differential forms, where no genericity hypotheses are required. We prove the consistency of our definition and the first properties of $\mathcal{Z}^f$ in Chapters 9, 10, and 11, after the needed study of the compactifications of the involved configuration spaces in Chapter 8.

To compute and use an invariant of links or manifolds, it is interesting to cut links or manifolds into elementary pieces and understand how one can recover the invariant from the invariants of the pieces. In the third part of this book, we achieve this task with elementary pieces that are *tangles* in rational homology cylinders as in Figure 1.12. These tangles are cobordisms between planar configurations of points. We define them precisely in Section 13.1. They may be composed in many ways, horizontally, vertically, and by insertions in tubular neighborhoods of other tangle representatives. Such insertions are called *cablings*. We generalize $\mathcal{Z}^f$ to tangles and describe the properties of our generalized $\mathcal{Z}^f$ under the mentioned compositions in the third part of this book.

As already mentioned, a fundamental property of $\mathcal{Z}^f$ is its universality among finite type invariants. For the restriction of $\mathcal{Z}^f$ to links in $\mathbb{R}^3$, universality refers to the Vassiliev theory of finite type invariants based on crossing changes. For the restriction of $\mathcal{Z}^f$ to rational homology spheres, universality refers to the Moussard theory based on rational Lagrangian-preserving surgeries. The proofs of universality involve computations of iterated discrete

<footnote n="15">¹⁵ *Framed links* or *parallelized links* are links equipped with a parallel (up to isotopy).</footnote>
<figure><img src="image_32.png" /><figcaption>Figure 1.12: A tangle representative in $D_1 \times [0, 1]$</figcaption></figure>

derivatives of $\mathcal{Z}^f$ in the same spirit as the Bar-Natan result recalled in the end of Subsection 1.2.5. The book’s fourth part presents these computations and some consequences for the general invariant $\mathcal{Z}^f$ of framed tangles in rational homology cylinders.

## 1.3 A quicker introduction

This quicker introduction, for experienced topologists, is independent of the first one. Beginners can read it after the warm-up of the slower one. Here, we describe the invariant $\mathcal{Z}$ of $n$-component links $L$ in rational homology 3-spheres $R$ studied in this book, more precisely. We also specify some notions vaguely introduced in the slow introduction of Section 1.2, and we say more on the mathematical landscape around $\mathcal{Z}$.

### 1.3.1 On the construction of $\mathcal{Z}$

The invariant $\mathcal{Z}(L) = (\mathcal{Z}_k(L))_{k \in \mathbb{N}}$ of an $n$-component link $L$ in a rational homology 3-sphere $R$ is valued in a graded space generated by uni-trivalent graphs. Its degree $k$ part is a sum

$$ \mathcal{Z}_k(L) = \sum_{\Gamma} \mathcal{Z}_{\Gamma}(L) [\Gamma], $$

running over such graphs $\Gamma$ with $2k$ vertices. The coefficient $\mathcal{Z}_{\Gamma}(L)$ “counts” embeddings of $\Gamma$ in $R$ mapping the univalent vertices of $\Gamma$ to $L$, in a sense explained in the book. Let us slightly specify that sense.
For technical reasons, we remove a point $\infty$ from our rational homology 3-spheres $R$ to transform them into open manifolds $\tilde{R}$. When $R$ is the standard sphere $S^3$, the punctured $\mathbb{Q}$-sphere $\tilde{R}$ is $\mathbb{R}^3$.

Let $\Delta(\tilde{R}^2)$ denote the diagonal of $\tilde{R}^2$. Following William Fulton, Robert MacPherson [FM94], Maxim Kontsevich [Kon94], Scott Axelrod, Isadore Singer [AS94, Section 5], and others, we will introduce a suitable smooth compactification $C_2(R)$ (with boundary and ridges) of $\tilde{R}^2 \setminus \Delta(\tilde{R}^2)$ such that the map

$$ p_{S^2}: \begin{array}{ccc} (\mathbb{R}^3)^2 \setminus \Delta((\mathbb{R}^3)^2) & \to & S^2 \\ (x, y) & \mapsto & \frac{1}{\|y-x\|}(y-x) \end{array} $$

extends to $C_2(S^3)$. We will introduce a notion of propagating chain and the dual notion of propagating form for $R$. When $R = S^3$, for any $X \in S^2$, the submanifold $p_{S^2}^{-1}(X)$ of $C_2(S^3)$ is an example of a propagating chain. For any 2-form $\omega_S$ on $S^2$ such that $\int_{S^2} \omega_S = 1$, the form $p_{S^2}^*(\omega_S)$ is an example of a propagating form. The propagating chain $p_{S^2}^{-1}(X)$ is a model propagating chain. The propagating form $p_{S^2}^*(\omega_S)$ is a model propagating form. In general, a propagating chain is a 4-dimensional rational chain (i.e., a finite rational combination of oriented compact 4-manifolds with possible ridges) of $C_2(R)$, while a propagating form is a closed 2-form on $C_2(R)$. Both have to satisfy some conditions on the boundary of $C_2(R)$, which make them share sufficiently many properties with our model propagating chains or forms.

In particular, for any propagating form $\omega$, for any propagating chain $P$, and for any two-component link $(J, K)$: $S^1 \sqcup S^1 \to \tilde{R}$, we have

$$ \int_{J \times K \subset C_2(R)} \omega = \langle J \times K, P \rangle_{C_2(R)} = lk(J, K), $$

where $\langle \cdot, \cdot \rangle_{C_2(R)}$ stands for the algebraic intersection in $C_2(R)$, and $lk$ is the linking number in $R$.

Propagating forms and propagating chains are both called propagators when their nature is clear from the context. The above equalities tell us in which way “propagators represent the linking form”.

A Jacobi diagram $\Gamma$ on $\bigsqcup_{i=1}^n S^1$ is a uni-trivalent graph $\Gamma$ equipped with an isotopy class of injections from its set $U(\Gamma)$ of univalent vertices into the domain $\bigsqcup_{i=1}^n S^1$ of a link $L$. In the figures, we represent the domain of $L$ by dashed circles and put the univalent vertices of $\Gamma$ on their images under an injection of the given isotopy class as in Figure 1.13. Let $V(\Gamma)$, $T(\Gamma)$, and $E(\Gamma)$ respectively denote the set of vertices, trivalent vertices, and edges of $\Gamma$. The configuration space $\tilde{C}(R, L; \Gamma)$ is the set of injections from $V(\Gamma)$ to $\tilde{R}$ that map the set $U(\Gamma)$ of univalent vertices of $\Gamma$ to $L$ and induce the given
<figure><img src="image_33.png" /><figcaption>Figure 1.13: A (plain) Jacobi diagram on (the dashed) $S^1$</figcaption></figure>

isotopy class of injections. It is an open submanifold of $\tilde{R}^{T(\Gamma)} \times L^{U(\Gamma)}$. Assume that $\Gamma$ has no *looped edge* like $\text{--}\circ$ and that its edges are oriented. Then each edge $e$ of $\Gamma$ provides a natural restriction map $p(\Gamma, e)$ from $\tilde{C}(R, L; \Gamma)$ to $\tilde{R}^2 \setminus \Delta(\tilde{R}^2)$. When propagating forms $\omega(e)$ are associated to the edges, this allows one to define a real number

$$I(R, L, \Gamma, (\omega(e))_e) = \int_{\tilde{C}(R, L; \Gamma)} \bigwedge_{e \in E(\Gamma)} p(\Gamma, e)^* (\omega(e)).$$

Similarly and dually, when propagating chains $P(e)$ in general position are associated to the edges, one can define a rational number $I(R, L, \Gamma, (P(e))_e)$ as the algebraic intersection of the codimension-two chains $p(\Gamma, e)^{-1}(P(e))$ in $\tilde{C}(R, L; \Gamma)$.

For example, the Jacobi diagram $\Gamma$ pictured as $i \prec \text{--} \succ \text{--} \succ \text{--} \succ j$, is an oriented edge $e$ from a univalent vertex that must go to the component $K_i$ of $L$ to another univalent vertex, which must go to another component $K_j$ of $L$. The associated configuration space $\tilde{C}(R, L; \Gamma)$ is $K_i \times K_j$, According to the given property of our propagators, we have

$$I(R, L, i \prec \text{--} \succ \text{--} \succ \text{--} \succ j, \omega(e)) = \int_{K_i \times K_j \subset C_2(R)} \omega(e) = lk(K_i, K_j)$$

and

$$I(R, L, i \prec \text{--} \succ \text{--} \succ \text{--} \succ j, P(e)) = \langle K_i \times K_j, P(e) \rangle_{C_2(R)} = lk(K_i, K_j)$$

for any propagating form $\omega(e)$ and for any propagating chain $P(e)$.

As another example, consider the Jacobi diagram $\ominus$ with two trivalent vertices and three edges $e_1, e_2$, and $e_3$ from one vertex to the other. When $R$ is a $\mathbb{Z}$-sphere, we will show how one can choose propagators $P(e_i)$ and $\omega(e_i)$ so that we have

$$I(R, \emptyset, \ominus, (\omega(e))_{e \in \{e_1, e_2, e_3\}}) = I(R, \emptyset, \ominus, (P(e))_{e \in \{e_1, e_2, e_3\}}) = 6\lambda_{CW}(R),$$

where $\lambda_{CW}$ is the Casson invariant normalized as in [AM90, GM92, Mar88]. We can choose the above propagating forms so that $\omega(e_1) = \omega(e_2) = \omega(e_3)$.
In particular, the Casson invariant, which may be written as

$$ \lambda_{CW}(R) = \frac{1}{6} \int_{C_2(R)} \omega(e_1)^3, $$

may be viewed as a “cube of the linking number”.

In general, when R is a Z-sphere, one can choose a propagating form ω for R and set ω(e) = ω for all edges e of Jacobi diagrams so that the real coefficient Z_Γ(L) in “Z(L) = Σ_Γ Z_Γ(L) [Γ]” is the product of I(R, L, Γ, (ω)_e) and a constant which depends only on Γ. The coefficient Z_Γ(L) can alternatively be obtained by averaging some I(R, L, Γ, (P(e))_e) over ways of equipping edges of Γ by propagating chains in a fixed set of generic propagating chains, and over ways of orienting the edges of Γ. The coefficients Z_Γ(L) depend on propagator choices, but relations among Jacobi diagrams in the target space A(∪_i=1 S^1) of Z—generated by Jacobi diagrams—ensure that Z(L) is an isotopy invariant. The invariant Z may be thought of as a series of higher-order linking invariants.

The definition of Z that is presented here is a generalization of the definition that I explained in detail for Q-spheres in the unsubmitted preprint [Les04a], which was inspired by [KT99] and discussions with Dylan Thurston in Kyoto in 2001.

The present definition also includes links in Q-spheres. Most of the additional arguments involved in the construction for links already appear in many places. We repeat them to make the book as self-contained as possible. We present many variants of the definitions and make them as flexible as possible because the flexibility has proved useful in many generalizations and applications of these constructions, such as equivariant constructions in [Les11, Les13], or the recent explicit computations of integrals over configuration spaces by David Leturcq [Let23, Let22], in addition to the applications presented in this book. David Leturcq obtained an expression of the Alexander polynomial of knots in Q-spheres in terms of such integrals. His expression identifies some combinations of integrals over configuration spaces with coefficients of the Alexander polynomial.

With our flexible definition of a propagating chain, there is a natural propagating chain associated to a generic Morse function on a punctured Q-sphere, and to a generic metric, as described in Subsection 1.2.8. The main part of such a propagator is the space of pairs of points on a gradient line such that the second point is after the first one. Greg Kuperberg and I constructed such a Morse propagator in [Les15a] for Morse functions without minima or maxima. Independent work of Tadayuki Watanabe [Wat18a] allows one to generalize these propagators to any Morse function. Thus, up to some corrections, Z counts embeddings of graphs whose edges embed in gradient
lines of Morse functions as in a Fukaya article [Fuk96]. Tadayuki Watanabe has used similar constructions in his recent construction of exotic elements in the fundamental group of the group of diffeomorphisms of $S^4$ [Wat18b]. This book contains a framework to study these questions precisely.<fnref n="16" />

We will also show how the construction of $\mathcal{Z}$ extends to tangles in rational homology cylinders so that $\mathcal{Z}$ extends to a functor $\mathcal{Z}^f$ on a category of framed tangles with many important properties. These properties of the functor $\mathcal{Z}^f$ are stated in Theorem 13.12, one of this book's main original theorems. They provide tools to reduce the computation of $\mathcal{Z}^f$ to its evaluation at elementary pieces.

### 1.3.2 More mathematical context

**Finite type invariants.** The finite type invariant concept for knots was introduced in the 90's in order to classify knot invariants, with the work of Victor Vassiliev, Mikhail Goussarov, and Dror Bar-Natan, shortly after the birth of numerous quantum knot invariants, described by Vladimir Turaev in [Tur10]. Tomotada Ohtsuki extended this very useful concept to 3-manifold invariants [Oht96]. See also [Oht02]. Theories of finite type invariants in dimension 3 are defined from a set $\mathcal{O}$ of operations on links or 3-manifolds. In the case of links in $\mathbb{R}^3$, $\mathcal{O}$ is the set $\mathcal{O}_V$ of crossing changes $\xrightleftharpoons{\sim} \xrightleftharpoons{\sim}$. The variation of an invariant $\lambda$ under an operation of $\mathcal{O}$ may be thought of as a discrete derivative. When $k$ independent operations $o_1, \dots, o_k$ on a pair $(R, L)$ consisting of a link $L$ in a $\mathbb{Q}$-sphere $R$ are given, for a part $I$ of $\{1, \dots, k\}$ with cardinality $|I|$, the pair $(R, L)((o_i)_{i \in I})$ is the pair obtained from $(R, L)$ by applying the operations $o_i$ for $i \in I$. Then the alternate sum

$$ \sum_{I \subseteq \{1, \dots, k\}} (-1)^{|I|} \lambda((R, L)((o_i)_{i \in I})) $$

may be thought of as the $k^{th}$ derivative of $\lambda$ with respect to $\{o_1, \dots, o_k\}$ at $(R, L)$. An invariant of degree at most $k$ with respect to $\mathcal{O}$ is an invariant whose degree $k+1$ derivatives vanish. A finite type invariant with respect to $\mathcal{O}$ is an invariant that is of degree at most $k$ for some positive integer $k$. Finite type invariants of links in $\mathbb{R}^3$ with respect to the set of crossing changes are called *Vassiliev invariants*.

In this case of links in $\mathbb{R}^3$, Daniel Altschüler and Laurent Freidel [AF97] proved that the invariant $\mathcal{Z}$ described in this book is a *universal Vassiliev invariant*, meaning that all real-valued Vassiliev invariants of links in $\mathbb{R}^3$

<footnote n="16">¹⁶We mainly develop the framework for tangles in dimension 3, but many arguments can be easily adapted in higher dimensions. See [Wat18b, Let21, Let23], for example.</footnote>
factor through $\mathcal{Z}$. Since all the quantum invariants of [Tur10] can be viewed
as sequences of finite type invariants, $\mathcal{Z}$ also contains all these invariants such
as the Jones polynomial, its colored versions, the HOMFLY polynomial...
Dylan Thurston proved similar universality results in [Thu99] independently.
He also showed that $\mathcal{Z}$ is rational. Further substantial work of Sylvain Poirier
in [Poi00] allowed me to identify the invariant $\mathcal{Z}$ with the famous Kontsevich
integral of links in $\mathbb{R}^3$—described in [BN95a], in [Oht02, Chapter 6], and in
[CDM12] by Sergei Chmutov, Sergei Duzhin, and Jacob Mostovoy—up to a
change of variables described in [Les02] in terms of an “anomaly”, which is
sometimes called the *Bott–Taubes anomaly*.

Let us now describe operations on 3-manifolds. The boundary $\partial A$ of a
genus $g$ Q-handlebody is a closed oriented genus $g$ surface. The *Lagrangian*
$\mathcal{L}_A$ of a compact 3-manifold $A$ is the kernel of the map induced by the
inclusion from $H_1(\partial A; \mathbb{Q})$ to $H_1(A; \mathbb{Q})$. (In Figure 1.1 of $H_g$, the Lagrangian
of $H_g$ is freely generated by the classes of the curves $a_i$.)

An *integral* (resp. *rational*) *Lagrangian-Preserving* (or *LP*) *surgery* ($A'/A$) is the replacement of an integral (resp. rational) homology handlebody $A$ embedded in the interior of a 3-manifold $M$ by another such $A'$ whose boundary $\partial A'$ is identified with $\partial A$ by an orientation-preserving diffeomorphism that sends $\mathcal{L}_{A'}$ to $\mathcal{L}_A$.

Theories of finite type invariants of integer (resp. rational) homology 3-
spheres $R$ can be defined from the set $O_{\mathcal{L}}^{\mathbb{Z}}$ (resp. $O_{\mathcal{L}}^{\mathbb{Q}}$) of integral (resp. ratio-
nal) LP-surgeries. For $\mathbb{Z}$-spheres, results of Kazuo Habiro [Hab00], Stavros
Garoufalidis, Mikhail Goussarov, and Michael Polyak [GGP01], and Em-
manuel Auclair and me [AL05] imply that the theory of real-valued finite
type invariants with respect to $O_{\mathcal{L}}^{\mathbb{Z}}$ is equivalent to the original theory de-
fined by Tomotada Ohtsuki in [Oht96] using surgeries on algebraically split
links. We will call this theory the *Ohtsuki–Goussarov–Habiro theory*.

Greg Kuperberg and Dylan Thurston first showed that the restriction of
$\mathcal{Z}$ to integer homology 3-spheres (equipped with empty links) is a *universal
finite type invariant* of $\mathbb{Z}$-spheres with respect to $O_{\mathcal{L}}^{\mathbb{Z}}$ in [KT99].

As in the case of links in $\mathbb{R}^3$, their proof of universality rests on a com-
putation of the $k^{th}$ derivatives of the degree $k$ part $\mathcal{Z}_k$ of the invariant
$\mathcal{Z} = (\mathcal{Z}_k)_{k \in \mathbb{Z}}$, which proves that $\mathcal{Z}_k$ is a degree $k$ invariant whose $k^{th}$ deriva-
tives are universal in the following sense. All the $k^{th}$ derivatives of degree $k$
real-valued invariants factor through them.

The “Universality part” of this book will be devoted to a general computation of the $k^{th}$ derivatives of the extension of $\mathcal{Z}_k$ to tangles with respect both to $O_V$ and $O_{\mathcal{L}}^{\mathbb{Q}}$ (which contains $O_{\mathcal{L}}^{\mathbb{Z}}$). The resulting formulae stated in Theorems 17.30 and 18.6 are crucial properties of $\mathcal{Z}$. Theorem 18.6 is one of the main original results of this book.
I first proved the splitting formulae, which compute the $k^{th}$ derivatives of the degree $k$ part $Z_k$ with respect to $O_L^Q$, in [Les04b] for the restriction of $Z_k$ to $\mathcal{Q}$-spheres. They allowed Delphine Moussard to classify finite type invariants of $\mathcal{Q}$-spheres with respect to $O_L^Q$ in [Mou12]. In particular, she proved that, when associated with the $p$-valuations of the cardinality $|H_1|$ of the torsion first homology group, $\mathcal{Z}$ is a *universal finite type invariant* of $\mathcal{Q}$-spheres with respect to $O_L^Q$. Together with results of Gwénaël Mas-suyeau [Mas14] who proved that the LMO invariant $Z_{LMO}$ of Thang Lê, Jun Murakami, and Tomotada Ohtsuki [LMO98] satisfies the same formulae, the Moussard classification implies that $\mathcal{Z}$ and $Z_{LMO}$ are equivalent in the sense that they distinguish the same $\mathcal{Q}$-spheres with identical $|H_1|$.

Thus, the invariant $\mathcal{Z}$ is as powerful as the famous LMO invariant for $\mathcal{Q}$-spheres, and as the famous Kontsevich integral for links. The Kontsevich integral $Z^K$ and its relations with the theory of quantum groups developed by Drinfeld and Jimbo have been extensively studied. See the books [Kas95, Oht02, CDM12], for instance. Thang Lê, Jun Murakami [LM96, Theorem 10], and Christian Kassel [Kas95, Theorem XX.8.3] independently showed how the Turaev quantum link invariants [Tur88], which include the HOMFLYPT and Kauffman polynomials, can be recovered from $Z^K$. See also [Oht02, Theorem 6.14]. An explicit way of recovering the Alexander polynomial from $Z^K$ for knots in $\mathbb{R}^3$ based on [BNG96] can be found in [CDM12, §11.2.4]. Tomotada Ohtsuki computed the two-loop part of the Kontsevich integral—which coincides with the two-loop part of $\mathcal{Z}$ according to Corollary 12.29 and Note 10.14—for the genus one knots [Oht07]. Dror Bar-Natan, Thang Lê, and Dylan Thurston computed the Kontsevich integral of the trivial knot $O$ [BNLT03].<sup>17</sup> Since $\mathcal{Z}$ is obtained from the Kontsevich integral by a change of variables determined by the anomaly as in Corollary 12.29, these results have direct corollaries for $\mathcal{Z}$.

The LMO invariant and its generalizations for links in $\mathcal{Q}$-spheres [LMO98, Oht02, BNGRT02a, BNGRT02b, BNGRT04] are defined from the Kontsevich integral of links in $\mathbb{R}^3$, in a combinatorial way. Any compact oriented 3-manifold can be presented by a *framed link* of $\mathbb{R}^3$, which is a link equipped with a favorite parallel, according to a theorem proved by Raymond Lickorish and Andrew Wallace, independently and nicely reproved by Colin Rourke in [Rou85]. The *Kirby moves* are specific modifications of framed links that do not change the presented manifold. According to a theorem of Robion Kirby, two framed links present the same manifold if and only if they are related by a finite sequence of Kirby moves. The LMO invariant of a 3-manifold is

<footnote n="17"><sup>17</sup>The computation for the torus knots follows as in [Les99, Proposition 3.16] or [CDM12, Section 9.3]. See also the computations of Julien Marché in [Mar04].</footnote>
defined from the Kontsevich integral of a framed link that presents such a
manifold. The proof of its invariance relies on the cited Kirby theorem. The
combinatorial nature of $Z_{LMO}$ allowed Dror Bar-Natan and Ruth Lawrence
to compute $Z_{LMO}$ for lens spaces and Seifert fibered spaces [BNL04].

When restricted to braids, the Kontsevich integral has a natural geo-
metric meaning. It measures how the strands turn around each other (see
[CDM12] or [Les99, Section 1]), and it defines morphisms from braid groups
to algebras of horizontal chord diagrams. Dror Bar-Natan extended the Kont-
sevich integral to links [BN95a]. Thang Lê and Jun Murakami extended the
Kontsevich integral to a functor from framed tangles to a category of Jacobi
diagrams [LM96]. Next, Thang Lê, Jun Murakami, and Tomotada Ohtsuki
defined the LMO invariant from the Le–Murakami–Kontsevich invariant of
surgery presentations of the 3-manifolds using tricky algebraic manipulations
of Jacobi diagrams with the help of Kirby calculus [LMO98, Oht02]. Though
some of the physical meaning of the LMO invariant can be recovered from
its universality properties, much of it gets lost in the manipulations.

The presented construction of $\mathcal{Z}$ is much more physical, geometric, and
natural—at least to me. It does not rely on the Kirby theorem and provides
information about graph embeddings in $\mathbb{Q}$-spheres. On the other hand, some
explicit computations or properties available for the LMO invariant have yet
to be performed or proved for $\mathcal{Z}$.

1.4 Book organization

Chapter 2 completes our slow introduction to the linking number of Sec-
tions 1.2.1 and 1.2.3. It contains more conventions and arguments used
throughout this book. Chapter 3 introduces our definitions of propagators.
These propagators are the basic ingredients of all our constructions. Most of
the time, they are associated with a parallelization of the 3-manifold. The
Theta invariant is the simplest 3-manifold invariant that can be derived from
the techniques described in this book. We present it in detail in Chapter 4.
We first describe $\Theta$ as an invariant of a parallelized punctured $\mathbb{Q}$-sphere
$(\tilde{R}, \tau)$. The invariant $\Theta(R, \tau)$ is the intersection of three propagating chains
associated with the given parallelization $\tau$ of $\tilde{R}$ in the two-point configuration
space $C_2(R)$. Equivalently, the invariant $\Theta(R, \tau)$ is the integral over $C_2(R)$
of the cube of a propagating form associated with $\tau$. Next, we transform
$\Theta$ to an invariant of $\mathbb{Q}$-spheres using relative Pontrjagin classes, also called
Hirzebruch defects, as Greg Kuperberg and Dylan Thurston did in [KT99].
Like $\Theta$, the series $\mathcal{Z}$ comes from a more natural invariant $\mathcal{Z}^f$ of parallelized
links $(L, L_\|)$ in parallelized punctured $\mathbb{Q}$-spheres, constructed with associ-
ated propagators. We transform $\mathcal{Z}^f(R, L, L_\parallel, \tau)$ to an invariant $\mathcal{Z}$ of links $L$ in $\mathcal{Q}$-spheres using a function of linking numbers associated to the link parallelizations, Pontrjagin numbers associated to the manifold parallelizations, and constants $\alpha$ and $\beta$ called *anomalies*. Chapter 5 presents parallelizations of oriented 3-manifolds with boundaries and associated Pontrjagin numbers in detail. It closes this introductory part.

Thus, the book's first part describes the degree one part of the graded invariant $\mathcal{Z}$ for links in $\mathcal{Q}$-spheres, which is determined by the linking numbers of the components and the $\Theta$-invariant of the ambient manifold. The book's second part is devoted to the general presentation of $\mathcal{Z}$ for links in $\mathcal{Q}$-spheres. In this part, we first review various theories of finite type invariants for which specializations of $\mathcal{Z}$ will be universal finite type invariants. This allows us to introduce the spaces of Jacobi diagrams in which $\mathcal{Z}$ takes its values, in a natural way, in Chapter 6. The complete definitions of $\mathcal{Z}$ for links in $\mathcal{Q}$-spheres are given in Chapter 7 without proofs of consistency. We show that these definitions make sense and do not depend on the involved choices of propagating forms in Chapters 9 and 10. Our proofs rely on the study of suitable compactifications of configuration spaces presented in Chapter 8 and on some standard arguments of the subject already appearing in many places starting with [Kon94], [BT94]... This second part of the book ends with discrete equivalent definitions of $\mathcal{Z}$ in terms of propagating chains and algebraic intersections rather than propagating forms and integrals in Chapter 11. These definitions make clear that the invariant $\mathcal{Z}$ is rational. The other main properties of $\mathcal{Z}$ are precisely described in Sections 10.1, 10.6, 13.3, and Chapter 18. Some of them involve the extension of $\mathcal{Z}$ to tangles, which can be found in Theorem 12.7.

The book's third part is devoted to this extension of $\mathcal{Z}$ to tangles, still denoted by $\mathcal{Z}$. We introduce the framed version $\mathcal{Z}^f$ of $\mathcal{Z}$ for framed tangles in Definition 12.12. The spirit of the definition is the same. However, its justification is more difficult because the involved compactified configuration spaces are more complicated. They are no longer smooth manifolds with ridges and have additional types of faces. We first present the definition and properties of the extension without proofs in Chapters 12 and 13. Next, we justify them in Chapters 14 and 17, respectively. In particular, Chapter 17 contains the proofs of many properties of the link invariant $\mathcal{Z}$. These proofs involve easy-to-discretize variants of the functor $\mathcal{Z}^f$, which are interesting on their own and presented in Section 16.2. The third part of the book ends with the computation of the iterated derivatives of the generalized $\mathcal{Z}$ with respect to crossing changes, in Section 17.6. This computation proves that the restriction of $\mathcal{Z}$ to links in $S^3$ is a universal Vassiliev invariant.

The fourth part focuses on the computation of the iterated derivatives
of the generalized $\mathcal{Z}$ with respect to rational LP-surgeries. It begins with Chapter 18, stating the main results and their corollaries, and reducing their proofs to the proofs of two key propositions, presented in Chapter 20. The proofs of these propositions involve the introduction of a more flexible definition of $\mathcal{Z}$ because the restriction of a parallelization of a $\mathbb{Q}$-sphere to the exterior of a $\mathbb{Q}$-handlebody does not necessarily extend to a $\mathbb{Q}$-handlebody that replaces the former one during a rational LP-surgery. Chapter 19 contains an extension of the notion of parallelization to a more flexible notion of *pseudo-parallelization* and a corresponding more flexible definition of $\mathcal{Z}$. Pseudo-parallelizations also have associated propagators and Pontrjagin numbers. They easily extend to arbitrary $\mathbb{Q}$-handlebodies. More flexible variants of the definition of $\mathcal{Z}$ based on pseudo-parallelizations can be found in Chapter 21.

The book ends with two appendices. Appendix A lists the basic results and techniques of algebraic topology used in the book. Appendix B reviews the used properties of differential forms and de Rham cohomology.

Most chapters have their own detailed introduction. Many cross-references help the reader choose what she/he wants to read.

## 1.5 Book genesis

At first, this book aimed at presenting the results of two preprints [Les04a, Les04b] and lecture notes [Les15b]. It contains generalizations of the results of these preprints to wider settings. I have never submitted the preprints [Les04a, Les04b] for publication. Edward Witten's insight into the perturbative expansion of the Chern-Simons theory [Wit89] inspired the mathematical guidelines for constructing the invariant $\mathcal{Z}$. Maxim Kontsevich gave them in [Kon94, Section 2]. Greg Kuperberg and Dylan Thurston developed these guidelines in [KT99]. They defined $\mathcal{Z}$ for $\mathbb{Q}$-spheres and sketched a proof that the restriction of $\mathcal{Z}$ to $\mathbb{Z}$-spheres is a universal finite type invariant of $\mathbb{Z}$-spheres in the Ohtsuki-Goussarov-Habiro sense. This allowed them to identify the degree one part of $\mathcal{Z}$ with the Casson invariant for $\mathbb{Z}$-spheres. I thank Dylan Thurston for explaining to me his joint work with Greg Kuperberg in Kyoto in 2001.

In [Les04b], I proved splitting formulae for $\mathcal{Z}$. These formulae compute derivatives of $\mathcal{Z}$ with respect to rational LP-surgeries. They generalize similar Kuperberg-Thurston implicit formulae about Torelli surgeries. These formulae allowed me to identify the degree one part of $\mathcal{Z}$ with the Walker generalization of the Casson invariant for $\mathbb{Q}$-spheres, in [Les04b, Section 6]. They also allowed Delphine Moussard to classify finite type invariants with
respect to these rational LP-surgeries and prove that all such real-valued fi-
nite type invariants factor through some “augmentation” of $\mathcal{Z}$ by invariants
derived from the order of the $H_1(.; \mathcal{Z})$, in [Mou12]. In [Mas14], Gwénaël
Massuyeau proved that the LMO invariant of Thang Lê, Jun Murakami, and
Tomotada Ohtsuki [LMO98] satisfies the same splitting formulae as $\mathcal{Z}$. Thus,
the Moussard classification implies that $\mathcal{Z}$ and $Z_{LMO}$ are equivalent in the
sense that they distinguish the same Q-spheres with identical $|H_1(.; \mathcal{Z})|$. In
order to write the proof of my splitting formulae, I needed to specify the
definition of $\mathcal{Z}$ and I described the Kontsevich–Kuperberg–Thurston con-
struction in detail in [Les04a].

In [Les15b], mixing known constructions in the case of links in $\mathbb{R}^3$ with the construction of $\mathcal{Z}$ allowed me to define a natural extension of $\mathcal{Z}$ as an invariant of links in Q-spheres. This extension also generalizes invariants of links in $\mathbb{R}^3$ defined by Enore Guadagnini, Maurizio Martellini, and Mihail Mintchev [GMM90], Dror Bar-Natan [BN95b], and by Raoul Bott and Clifford Taubes [BT94], which emerged after the Witten work [Wit89].<sup>18</sup> I also gave more flexible definitions of $\mathcal{Z}$.

In addition to the revisited contents of the preprints [Les04a, Les04b] and
of the notes [Les15b], this book contains an extension of $\mathcal{Z}$ as a functorial
invariant of tangles in rational homology cylinders. It also contains the proofs
of many properties of this extension, which imply simpler properties for $\mathcal{Z}$,
such as the multiplicativity of $\mathcal{Z}$ under connected sum, for example. This
functorial extension, which generalizes the Poirier extension in [Poi00], and
its properties are new. They appear only in this book (to my knowledge).

Most of the properties of $\mathcal{Z}$ are very intuitive and rather easy to accept
after some hand-waving. Writing complete proofs is often more complicated
than one would expect. I hope I have succeeded in this task, which was
sometimes much more difficult than I expected.

1.6 Some open questions

1. A Vassiliev invariant is *odd* if it distinguishes some knot from the same knot with the opposite orientation. Are there odd Vassiliev invariants?

2. More generally, do Vassiliev invariants distinguish knots in $S^3$? In [Kup96], Greg Kuperberg proved that if they distinguish unoriented knots in $S^3$, then there exist odd Vassiliev invariants.

<footnote n="18"><sup>18</sup>The relation between the perturbative expansion of the Chern–Simons theory of the Witten article and the configuration space integral viewpoint is explained by Michael Polyak in [Pol05] and by Justin Sawon in [Saw06]. See also [Oht02, Appendix F]</footnote>
3. According to a theorem of Dror Bar-Natan and Ruth Lawrence [BNL04], the LMO invariant fails to distinguish rational homology spheres with isomorphic $H_1$. So, according to a Moussard theorem [Mou12], rational finite type invariants fail to distinguish $\mathbb{Q}$-spheres. Do finite type invariants distinguish $\mathbb{Z}$-spheres?

4. Compute the anomalies $\alpha$ and $\beta$ of Sections 10.3 and 10.2. For links in $\mathbb{R}^3$, I expressed the invariant $\mathcal{Z}$ as a function of the Kontsevich integral $Z^K$, described in [BN95a, Oht02, CDM12], and of the Bott-Taubes anomaly $\alpha$, in [Les02]. The computation of $\alpha$ would finish clarifying the relationship between $\mathcal{Z}$ and $Z^K$ for links in $\mathbb{R}^3$. See Note 10.14.

5. Find surgery formulae for $\mathcal{Z}$. Do the surgery formulae that define $Z_{LMO}$ from $Z^K$ define $\mathcal{Z}$ from its restriction to links in $\mathbb{R}^3$?

6. Compare $\mathcal{Z}$ with the LMO invariant $Z_{LMO}$ of Thang Lê, Jun Murakami, and Tomotada Ohtsuki described in [Oht02, Chapter 10].

7. Find relationships between $\mathcal{Z}$ or other finite type invariants and Heegaard Floer homologies. Recall the propagators associated to Heegaard diagrams of [Les15a] from Subsection 1.2.8.

8. Andrew Kricker defined a lift $\tilde{Z}^K$ of the Kontsevich integral $Z^K$ (or the LMO invariant) for null-homologous knots in $\mathbb{Q}$-spheres [Kri00, GK04]. The Kricker lift is valued in a space $\tilde{A}$ of trivalent diagrams whose edges are decorated by rational functions whose denominators divide the Alexander polynomial. Compare the Kricker lift $\tilde{Z}^K$ with the equivariant configuration space invariant $\tilde{Z}^c$ of [Les11, Les13] valued in the same diagram space $\tilde{A}$.

9. Does one obtain $\mathcal{Z}$ from $\tilde{Z}^c$ in the same way as one obtains $Z^K$ from $\tilde{Z}^K$?

10. Study extensions of $\mathcal{Z}$ to manifolds with boundary. Dorin Cheptea, Kazuo Habiro, and Gwénaël Massuyeau introduced a functorial extension of the LMO invariant in [CHM08]. See also [HM21].
# Chapter 2

## More on manifolds and on the
linking number

The first section of this chapter specifies some basic notions of differential topology, quickly and sometimes vaguely introduced in Subsection 1.2.2. It also contains some additional notation and conventions. The second section completes our discussion about the linking number of Sections 1.2.1 and 1.2.3.

### 2.1 More background material on manifolds

#### 2.1.1 Manifolds without boundary

This section presents a quick review of the notions of manifold and tangent bundle. We refer the reader to [Hir94, Chapter 1] by Morris Hirsch for a clean and complete introduction.

A topological $n$-dimensional manifold $M$ without boundary is a Hausdorff topological space that is a union of open subsets $U_i$ labeled in a countable set $I$ ($i \in I$), where every $U_i$ is identified with an open subset $V_i$ of $\mathbb{R}^n$ by a homeomorphism $\phi_i: U_i \to V_i$, called a *chart*. Such a collection $(\phi_i: U_i \to V_i)_{i \in I}$ of charts, for which $\cup_{i \in I} U_i = M$, is called an *atlas* of $M$. We consider manifolds up to homeomorphism. So homeomorphic manifolds are considered identical.

For $r = 0, \dots, \infty$, the topological manifold $M$ has a $C^r$-structure (induced by the atlas $(\phi_i)_{i \in I}$) or is a $C^r$-manifold, if, for each pair $\{i, j\} \subset I$, the transition map $\phi_j \circ \phi_i^{-1}$ defined on $\phi_i(U_i \cap U_j)$ is a $C^r$-diffeomorphism onto its image. The notion of $C^s$-maps, $s \le r$, from such a manifold to another one can be induced naturally from the known case for which the manifolds are open subsets of some $\mathbb{R}^n$, thanks to the local identifications provided by
the charts. Manifolds of class $C^r$ are considered up to $C^r$-diffeomorphism. They are called $C^r$-manifolds. Smooth manifolds are $C^\infty$-manifolds.

A $C^r$ embedding from a $C^r$ manifold $A$ into a $C^r$ manifold $M$ is an injective $C^r$ map $j: A \hookrightarrow M$ such that, for any point $a$ of $A$, there exist

* a $C^r$ diffeomorphism $\phi$ from an open neighborhood $U$ of $j(a)$ in $M$ to an open subset $V$ of $\mathbb{R}^n$ and

* an open neighborhood $U_A$ of $a$ in $A$ such that the restriction $j|_{U_A}$ of $j$ to $U_A$ is a $C^r$-diffeomorphism onto its image, which may be written as
  $$j(U_A) = j(A) \cap U = \phi^{-1}(V \cap (\mathbb{R}^d \times \{(0, \dots, 0)\})).$$

A submanifold of a manifold $M$ is the image of such an embedding into $M$.

The tangent space $T_x A$ to a $C^r$ submanifold $A$ of $\mathbb{R}^n$ at a point $x$ of $A$, for $r \ge 1$ is the vector space of all tangent vectors to a curve (i.e., a 1-dimensional submanifold) of $A$ at $x$. A well-known theorem [Hir94, Theorem 3.4, Chapter 1] asserts that any compact $C^r$-manifold, for $r \ge 1$ may be embedded in some $\mathbb{R}^d$, and thus viewed as a submanifold of $\mathbb{R}^d$. The tangent bundle $TA$ to $A$ is the union over the elements $x$ of $A$ of the $T_x A$. Its bundle projection $p: TA \to A$ maps an element $v$ of $T_x A$ to $x$. The tangent bundle to $\mathbb{R}^n$ is canonically isomorphic to $\mathbb{R}^n \times \mathbb{R}^n$. A $C^r$ diffeomorphism between two open sets of $\mathbb{R}^n$, together with its (first order) derivatives induces a canonical $C^{r-1}$-diffeomorphism between their tangent bundles. The notion of tangent bundle of any $C^r$ $n$-manifold, for $r \ge 1$, is naturally induced from the local identifications provided by the charts. A $C^r$ map $f$ from a $C^r$-manifold $M$ to another one $N$ has a well-defined tangent map, which is a map $Tf: TM \to TN$ restricting as a linear map $T_x f: T_x M \to T_{f(x)} N$ for any $x$ of $M$.

### 2.1.2 More on low-dimensional manifolds

We now review classical results, which ensure that for $n = 1, 2$ or $3$, any topological $n$-manifold may be equipped with a unique smooth (i.e., $C^\infty$) structure (up to diffeomorphism).

A topological manifold $M$ as in the previous section has a piecewise linear (or PL) structure (induced by the atlas $(\phi_i)_{i \in I}$) or is a PL-manifold, if, for each pair $\{i, j\} \subset I$, the transition map $\phi_{i,j} \circ \phi_i^{-1}$ is a piecewise linear homeomorphism onto its image. PL-manifolds are considered up to PL-homeomorphism.

An $n$-dimensional simplex is the convex hull of $(n+1)$ points that are not contained in an affine subspace of dimension $(n-1)$ in some $\mathbb{R}^k$, with $k \ge n$.
For example, a 1-dimensional simplex is a closed interval, a 2-dimensional simplex is a solid triangle, and a 3-dimensional simplex is a solid tetrahedron. A topological space $X$ has a triangulation, if it is a locally finite union of $k$-simplices (closed in $X$), which are the simplices of the triangulation, such that

• the simplices are embedded in $X$,

• every face of a simplex of the triangulation is a simplex of the triangulation,

• when two simplices of the triangulation are not disjoint, their intersection is a simplex of the triangulation.<fnref n="1" />

PL manifolds always have such triangulations.

When $n \le 3$, the above notion of PL-manifold coincides with the notions of smooth and topological manifold, according to the following theorem. This is no longer true when $n > 3$. See [Kui99] by Nicolaas Kuiper.

**Theorem 2.1.** Let $n$ be a natural integer such that $n \le 3$. Any topological $n$-manifold has a unique PL structure (up to PL homeomorphism). For any $r \in (\mathbb{N} \setminus \{0\}) \cup \{\infty\}$, any topological $n$-manifold has a unique $C^r$-structure (up to $C^r$ diffeomorphism).

This statement contains several theorems (see [Kui99]): The fact that any $C^1$-manifold in any dimension has a unique $C^r$-structure (up to $C^r$-diffeomorphism), for any $r \in (\mathbb{N} \setminus \{0, 1\}) \cup \{\infty\}$, follows from work of Hassler Whitney in 1936 [Whi36]. In 1934, Stewart Cairns [Cai35] equipped any $C^1$-manifold with a PL-structure. He proved that any PL manifold of dimension 3 arises as the image of a $C^1$-manifold under such a process [Cai40, Theorem III] in 1940. (This was already known in dimension less than 3.) Edwin Moise [Moi52] proved that any topological 3-manifold has a unique PL structure in 1952. (This was also already known in dimension less than 3.) In dimension 3, James Munkres [Mun60, Theorem 6.3] and Henry Whitehead [Whi61] completed this scheme in 1960 by their independent proofs of the uniqueness of a $C^1$-structure for any topological 3-manifold.

### 2.1.3 Connected sum

Let $M_1$ and $M_2$ be two smooth closed manifolds of dimension $n$. The connected sum $M_1\#M_2$ of $M_1$ and $M_2$ is defined as follows. For $i \in \{1, 2\}$, let

<footnote n="1">¹A collection of sets in $X$ is locally finite if each point of $X$ has a neighborhood in $X$ that intersects finitely many sets of the collection.</footnote>
$\phi_i: 2\mathring{B}^n \hookrightarrow M_i$ be a smooth embedding of the open ball $2\mathring{B}^n$ of radius 2 (centered at the origin) of the Euclidean vector space $\mathbb{R}^n$ into $M_i$, such that $\phi_1$ is orientation-preserving and $\phi_2$ is orientation-reversing. The elements of $(2\mathring{B}^n \setminus \{0\})$ may be written as $\lambda x$ for a unique pair $(\lambda, x) \in ]0, 2[ \times S^{n-1}$, where $S^{n-1}$ is the unit sphere of $\mathbb{R}^n$. Let $h: \phi_1(2\mathring{B}^n \setminus \{0\}) \to \phi_2(2\mathring{B}^n \setminus \{0\})$ be the diffeomorphism such that $h(\phi_1(\lambda x)) = \phi_2((2 - \lambda)x)$ for any $(\lambda, x) \in ]0, 2[ \times S^{n-1}$.

Then

$$M_1\#M_2 = (M_1 \setminus \{\phi_1(0)\}) \cup_h (M_2 \setminus \{\phi_2(0)\})$$

is the quotient space of $(M_1 \setminus \{\phi_1(0)\}) \sqcup (M_2 \setminus \{\phi_2(0)\})$, in which an element of $\phi_1(2\mathring{B}^n \setminus \{0\})$ is identified with its image under $h$. As a topological manifold, $M_1\#M_2$ can be written as

$$M_1\#M_2 = (M_1 \setminus \phi_1(\mathring{B}^n)) \cup_{\phi_1(S^{n-1})} h \phi_2(S^{n-1}) (M_2 \setminus \phi_2(\mathring{B}^n)).$$

### 2.1.4 Manifolds with boundary and ridges

A topological $n$-dimensional manifold $M$ with possible boundary is a Hausdorff topological space that is a union of open subsets $U_i$ labeled in a set $I$, ($i \in I$), where every $U_i$ is identified with an open subset $V_i$ of $]-\infty, 0]^k \times \mathbb{R}^{n-k}$ by a chart $\phi_i: U_i \to V_i$. The boundary of $]-\infty, 0]^k \times \mathbb{R}^{n-k}$ consists of the points $(x_1, \dots, x_n)$ of $]-\infty, 0]^k \times \mathbb{R}^{n-k}$ for which there exists $i \le k$ such that $x_i = 0$. The boundary of $M$ consists of the points mapped to the boundary of $]-\infty, 0]^k \times \mathbb{R}^{n-k}$ by a chart.

A map from an open subset $O$ of $]-\infty, 0]^k \times \mathbb{R}^{n-k}$ to an open subset of $]-\infty, 0]^{k'} \times \mathbb{R}^{n'-k'}$ is smooth (resp. $C^r$) at a point $x \in O$ if it extends as a smooth (resp. $C^r$) map from an open neighborhood of $x$ in $\mathbb{R}^n$ to $\mathbb{R}^{n'}$.

The topological manifold $M$ is a smooth manifold with ridges, if, for each pair $\{i, j\} \subset I$, the map $\phi_j \circ \phi_i^{-1}$ defined on $\phi_i(U_i \cap U_j)$ is a smooth diffeomorphism onto its image.<fnref n="2" /> When $k \le 1$ for any $i$, we simply say that $M$ is a smooth manifold with boundary. The codimension $j$ boundary of such a manifold $M$, which is denoted by $\partial_j(M)$, consists of the points that are mapped to points $(x_1, \dots, x_n)$ of $]-\infty, 0]^k \times \mathbb{R}^{n-k}$ for which there are at least $j$ indices $i \le k$ such that $x_i = 0$. It is a closed subset of $M$. We have $\partial M = \partial_1(M)$.

The codimension $j$ faces of such a smooth manifold $M$ with ridges are the connected components of $\partial_j(M) \setminus \partial_{j+1}(M)$. They are smooth manifolds of dimension $(n - j)$. The interior of $M$ is $M \setminus \partial M$.

For $j \ge 2$, the codimension $j$ faces are called ridges of $M$.

<footnote n="2">²Manifolds with ridges are often called manifolds with corners.</footnote>
### 2.1.5 Algebraic intersections

We start this subsection with more orientation conventions and notation. Again, unless otherwise mentioned, manifolds are smooth, compact, and oriented. The normal bundle to a submanifold $A$ in a manifold $M$, at a point $x$, is the quotient $T_x M/T_x A$ of tangent bundles at $x$. We denote it by $N_x A$ or $N_x(A)$. We orient $N_x A$ so that (a lift of an oriented basis of) $N_x A$ followed by (an oriented basis of) $T_x A$ induce the orientation of $T_x M$. The orientation of $N_x(A)$ is a coorientation of $A$ at $x$. The regular preimage of a submanifold under a map $f$ is oriented so that $f$ preserves the coorientations.

Two submanifolds $A$ and $B$ in a manifold $M$ are transverse if we have $T_x M = T_x A + T_x B$ at each $x \in A \cap B$. As proved in [Hir94, Chapter 3 (Theorem 2.4 in particular)], transversality is a generic condition. The intersection $A \cap B$ of two transverse submanifolds $A$ and $B$ in a manifold $M$ is a manifold. We orient $A \cap B$ so that the normal bundle to $A \cap B$ is $(N(A) \oplus N(B))$, fiberwise. In order to give a meaning to the sum $(N_x(A) \oplus N_x(B))$ at $x \in A \cap B$, pick a Riemannian metric on $M$. Such a metric identifies $N_x(A)$ with $T_x(A)^\perp$, $N_x(B)$ with $T_x(B)^\perp$, and $N_x(A \cap B)$ with $T_x(A \cap B)^\perp = T_x(A)^\perp \oplus T_x(B)^\perp$. Since the space of Riemannian metrics on $M$ is convex, and therefore connected, the induced orientation of $T_x(A \cap B)$ does not depend on the choice of a Riemannian metric.

Let $A, B, C$ be three pairwise transverse submanifolds in a manifold $M$ such that $A \cap B$ is transverse to $C$. The oriented intersection $(A \cap B) \cap C$ is a well-defined manifold. Our assumptions imply that at any $x \in A \cap B \cap C$, the sum $(T_x A)^\perp + (T_x B)^\perp + (T_x C)^\perp$ is a direct sum $(T_x A)^\perp \oplus (T_x B)^\perp \oplus (T_x C)^\perp$ for any Riemannian metric on $M$. So $A$ is also transverse to $B \cap C$, and $(A \cap B) \cap C = A \cap (B \cap C)$. Thus, the intersection of transverse, oriented submanifolds is a well-defined associative operation, where transverse submanifolds are manifolds such that the elementary pairwise intermediate possible intersections are well-defined, as above. This intersection is also commutative when the codimensions of the submanifolds are even.

Recall from Subsection 1.2.7 that, for two transverse submanifolds $A$ and $B$ of complementary dimensions in a manifold $M$, the sign $\pm 1$ of a point $x \in A \cap B$ is $+1$ if and only if $T_x M = T_x A \oplus T_x B$ as oriented vector spaces. This is equivalent to the condition that the orientation of the normal bundle to $x \in A \cap B$ coincides with the orientation of the ambient space $M$, that is that $T_x M = N_x A \oplus N_x B$ (as oriented vector spaces again—exercise). If $A$ and $B$ are of complementary dimensions in $M$, and if $A \cap B$ if finite, then the algebraic intersection $\langle A, B \rangle_M$ of $A$ and $B$ is the sum of the signs of the intersection points.

In a manifold $M$, a $k$-dimensional chain (resp. a $k$-dimensional rational
chain) is a finite combination with coefficients in $\mathbb{Z}$ (resp. in $\mathbb{Q}$) of (smooth, compact, oriented) $k$-dimensional submanifolds $C$ of $M$ with boundary and ridges, up to the identification of $(-1)C$ with $(-C)$ and other natural identifications (e.g., a $k$-manifold $A \cup B$ such that $A \cap B$ is a $(k-1)$-submanifold of $A \cup B$ is identified with the chain $A + B$). The boundary $\partial$ of chains is the linear map that maps a submanifold to its oriented boundary (with respect to the usual outward normal first convention). This boundary is the sum of the closures of the codimension-one faces when there are ridges. The canonical orientation of a point is the sign $+1$. So $\partial[0, 1] = \{1\} - \{0\}$. A $k$-dimensional chain whose boundary vanishes is a $k$-dimensional cycle, or a $k$-cycle for short.

If $A_1, \dots, A_k$ are $k$ transverse compact submanifolds of $M$ whose codimension sum is the dimension of $M$, then their algebraic intersection is defined to be $\langle A_1, \dots, A_k \rangle_M = \langle \cap_{i=1}^{k-1} A_i, A_k \rangle_M$. If $M$ is a connected manifold, which contains a point $x$, then the class of a 0-cycle in $H_0(M; \mathbb{Q}) = \mathbb{Q}[x] = \mathbb{Q}$ is a well-defined number. The algebraic intersection $\langle A_1, \dots, A_k \rangle_M$ can be equivalently defined to be the homology class of the (oriented) intersection $\cap_{i=1}^k A_i$. This algebraic intersection extends multilinearly to rational chains.

We will use the following lemma in Subsection 2.2.1.

**Lemma 2.2.** Let $A$ and $B$ be two transverse submanifolds of a $d$-dimensional manifold $M$ with disjoint boundaries. Let $\beta$ denote the dimension of $B$. Then

$$ \partial(A \cap B) = (-1)^{d-\beta} \partial A \cap B + A \cap \partial B. $$

**PROOF:** Note that $\partial(A \cap B) \subset \partial A \cup \partial B$. At a point $a \in \partial A$, $T_a M$ is oriented by $(N_a A, o, T_a \partial A)$, where $o$ is the outward normal to $A$. If $a \in \partial A \cap B$, then $o$ is also an outward normal for $A \cap B$, and $\partial(A \cap B)$ is cooriented by $(N_a A, N_a B, o)$, while $\partial A \cap B$ is cooriented by $(N_a A, o, N_a B)$. At a point $b \in A \cap \partial B$, both $\partial(A \cap B)$ and $A \cap \partial B$ are cooriented by $(N_b A, N_b B, o)$. $\square$

### 2.1.6 More on the degree

Here, we make the notion of walls in Subsection 1.2.5 more precise, by stating a lemma on the general behavior of the degree. We prove it with the Morse–Sard theorem 1.4. The formula of the lemma could also be justified with Stokes’ theorem.

**Lemma 2.3.** Let $n \in \mathbb{N}$. Let $M$ be a compact (oriented) $n$-manifold with possible boundary. Let $N$ be a connected (oriented) $n$-manifold. Let $f: M \to N$ be a smooth map. Let $a$ and $b$ be two distinct regular values of $f$ in the interior $\text{Int}(N)$ of $N$.
Then there exists an embedding $\gamma: [0, 1] \to \text{Int}(N)$ such that $\gamma(0) = a$, $\gamma(1) = b$, and, for any $x \in \partial M \cap f^{-1}(\gamma([0, 1]))$,

• $x$ is in an open face of $\partial M$ (of codimension one in $M$) and

• $T_{f(x)}N$ equals $\eta(x)T_{f(x)}\gamma \oplus T_xf(T_x\partial M)$ as an oriented vector space for some $\eta(x) = \pm 1$.

For any such embedding $\gamma$, we have

$$ \deg_a f - \deg_b f = \langle \gamma, f(\partial M) \rangle_N = \sum_{x \in f^{-1}(\gamma) \cap \partial M} \eta(x). $$

<figure><img src="image_34.png" /><figcaption>Figure 2.1: Lemma 2.3 for an embedding $f$ from $M$ to a rectangle $N$</figcaption></figure>

PROOF: Let $B^{n-1}$ be the unit ball of $\mathbb{R}^{n-1}$. Since $N$ is connected, there exists an (orientation-preserving) embedding $\Psi: B^{n-1} \times [-1, 2] \to \text{Int}(N)$ such that $\Psi(0, 0) = a$, $\Psi(0, 1) = b$, and $\Psi(\varepsilon B^{n-1} \times [-\varepsilon, \varepsilon])$ and $\Psi(\varepsilon B^{n-1} \times [1 - \varepsilon, 1 + \varepsilon])$ consist of regular values of $f$ for some $\varepsilon \in ]0, \frac{1}{100}[$.

If all the elements of $\Psi(\{0\} \times [0, 1])$ are regular values of $f$, then the preimage $f^{-1}(\Psi(\{0\} \times [0, 1]))$ is a disjoint union of intervals between a point of $f^{-1}(a)$ and a point of $f^{-1}(b)$, along which the sign of $\det(T_xf)$ is constant, and $\deg_a f$ is equal to $\deg_b f$. This proves that the degree is constant on any ball of regular values. Let us return to the general case and try to make a similar argument work. Set $N_\varepsilon = \varepsilon \dot{B}^{n-1} \times [0, 1]$, $M_\varepsilon = f^{-1}(\Psi(N_\varepsilon))$, and $f_\varepsilon = \Psi^{-1} \circ f|_{M_\varepsilon}$. For any $c \in \Psi(\dot{N}_\varepsilon)$, $\deg_c f = \deg_{\Psi^{-1}(c)}(f_\varepsilon)$.

Let $p_B$ and $p_I$ respectively denote the natural projections of $\varepsilon \dot{B}^{n-1} \times [0, 1]$ to its factors $\varepsilon \dot{B}^{n-1}$ and $[0, 1]$. The Morse–Sard theorem 1.4 guarantees the existence of a regular value $y$ of $p_B \circ f_\varepsilon$ in $\varepsilon \dot{B}^{n-1}$. Let $\gamma_y: [0, 1] \to N_\varepsilon$ map $t$ to $(y, t)$. Then $\gamma_y$ is cooriented by $p_B$ (via $T_w p_B: T_w N_\varepsilon \to \mathbb{R}^{n-1}$). So its preimage

$$ f_{\varepsilon}^{-1}(\gamma_y) = f_{\varepsilon}^{-1}(\gamma_y([0, 1])) = (p_B \circ f_{\varepsilon})^{-1}(y) $$

is a submanifold of dimension 1 of $M$, cooriented by $p_B \circ f_\varepsilon$. Let us show that the oriented boundary of $f_\varepsilon^{-1}(\gamma_y)$ is

$$ \partial f_{\varepsilon}^{-1}(\gamma_y) = -f_{\varepsilon}^{-1}((0, y)) + f_{\varepsilon}^{-1}((1, y)) + \sum_{x \in f_{\varepsilon}^{-1}(\gamma_y) \cap \partial M} \eta(x)x. $$
For a regular point $x$ of $f_\varepsilon$ (i.e., a point such that $f_\varepsilon$ is a local diffeomorphism near $x$), let $\delta(x) = \pm 1$ denote the sign of the Jacobian determinant of $f_\varepsilon$ at $x$. Near such a regular point $x$ in $(p_B \circ f_\varepsilon)^{-1}(y)$, the curve $\delta(x)f_\varepsilon^{-1}(\gamma_y)$ is oriented by $p_I \circ f_\varepsilon$. This shows that the above signs before $f_\varepsilon^{-1}((0, y))$ and $f_\varepsilon^{-1}((1, y))$ are correct.

Let us now check the sign associated to $x \in f_\varepsilon^{-1}(\gamma_y) \cap \partial M$. Let $N_{x,\partial M}$ denote the outward normal to $M$ at $x$. So we have $T_x M = \mathbb{R}N_{x,\partial M} \oplus T_x \partial M$ and

$$T_{f_\varepsilon(x)}N_\varepsilon = \eta(x)T_{f_\varepsilon(x)}\gamma_y \oplus T_x f_\varepsilon(T_x \partial M) = \delta(x)(T_x f_\varepsilon(\mathbb{R}N_{x,\partial M}) \oplus T_x f_\varepsilon(T_x \partial M)).$$

Therefore, the outward normal $N_{x,\partial M}$ to $\partial M$ is oriented by $\eta(x)\delta(x)p_I \circ f_\varepsilon$, and the sign of the scalar product of $T_x(f_\varepsilon^{-1}(\gamma_y))$ and $N_{x,\partial M}$ is $\eta(x)$.

Writing that the homology class of $\partial f_\varepsilon^{-1}(\gamma_y)$ is zero in $H_0(M)$ shows

$$\deg_{(0,y)} f_\varepsilon - \deg_{(1,y)} f_\varepsilon = \sum_{x \in f_\varepsilon^{-1}(\gamma_y) \cap \partial M} \eta(x).$$

This proves that the lemma holds for $\Psi(a_y = \gamma_y(0))$, $\Psi(b_y = \gamma_y(1))$, and $\Psi \circ \gamma_y$. Since $\Psi(a_y)$ and $a$ are in a common open ball $B_a$ of regular values of $f$ (in $\text{Int}(N) \setminus f(\partial M)$), and since $\Psi(b_y)$ and $b$ are in another common ball of regular values of $f$, we may modify $\Psi \circ \gamma_y$ to a smooth path $\gamma$ from $a$ to $b$ with all the properties of the statement.

Let us finish by proving that for any smooth embedded path $\gamma$ from $a$ to $b$ transverse to $f(\partial M)$, we have $\deg_a f - \deg_b f = \langle \gamma, f(\partial M) \rangle_N$. For any such path $\gamma$, we can construct a neighborhood embedding $\Psi$ of $\gamma$ as above, such that $\Psi(0, t) = \gamma(t)$ for all $t \in [0, 1]$, $\Psi(y \times ]-\varepsilon, 1+\varepsilon[)$ is transverse to $f(\partial M)$ for any $y \in \varepsilon\hat{B}^{n-1}$, and $\langle \gamma_y, f(\partial M) \rangle$ does not depend on $y \in \varepsilon\hat{B}^{n-1}$. So the previous study allows us to conclude. $\square$

## 2.2 On the linking number, again

### 2.2.1 A general definition of the linking number

**Lemma 2.4.** Let $J$ and $K$ be two rationally null-homologous disjoint cycles of respective dimensions $j$ and $k$ in a $d$-manifold $M$, where $d = j + k + 1$. There exists a rational $(j+1)$-chain $\Sigma_J$ bounded by $J$ transverse to $K$ and a rational $(k+1)$-chain $\Sigma_K$ bounded by $K$ transverse to $J$. For any two such rational chains $\Sigma_J$ and $\Sigma_K$, we have $\langle J, \Sigma_K \rangle_M = (-1)^{j+1}\langle \Sigma_J, K \rangle_M$. In particular, $\langle J, \Sigma_K \rangle_M$ is a topological invariant of $(J, K)$. It is denoted by $lk(J, K)$ and called the linking number of $J$ and $K$. We have

$$lk(J, K) = (-1)^{(j+1)(k+1)}lk(K, J).$$
PROOF: Since $K$ is rationally null-homologous, $K$ bounds a rational $(k+1)$-chain $\Sigma_K$. Without loss of generality, $\Sigma_K$ is assumed to be transverse to $\Sigma_J$, so $\Sigma_J \cap \Sigma_K$ is a rational 1-chain (which is a rational combination of circles and intervals). According to Lemma 2.2, we have

$$ \partial(\Sigma_J \cap \Sigma_K) = (-1)^{d+k+1} J \cap \Sigma_K + \Sigma_J \cap K. $$

Furthermore, the sum of the coefficients of the points in the left-hand side must be zero since this sum vanishes for the boundary of an interval. This proves $\langle J, \Sigma_K \rangle_M = (-1)^{d+k} \langle \Sigma_J, K \rangle_M$. Therefore, this rational number is independent of the chosen $\Sigma_J$ and $\Sigma_K$. Since we have

$$ (-1)^{d+k} \langle \Sigma_J, K \rangle_M = (-1)^{j+1} (-1)^{k(j+1)} \langle K, \Sigma_J \rangle_M, $$

$$ \text{we get } lk(J, K) = (-1)^{(j+1)(k+1)} lk(K, J). \qquad \square $$

**Remark 2.5.** Our sign convention for the linking number differs from that in [ST80, Section 77, page 288], where the linking number of cycles $J$ and $K$ as in the lemma is defined as $\langle \Sigma_J, K \rangle_M$, instead. The reason for our sign convention is justified in Remark 2.10.

In particular, the linking number of two rationally null-homologous disjoint links $J$ and $K$ in a 3-manifold $M$ is the algebraic intersection of a rational chain bounded by one of the links and the other.

For $K = \mathbb{Z}$ or $\mathbb{Q}$, any knot is rationally null-homologous in a $K$-sphere or in a $K$-ball (defined in Section 1.1). So the linking number of two disjoint knots always makes sense in such a 3-manifold.

A meridian $m_K$ of a knot $K$ is the (oriented) boundary of a disk that intersects $K$ once with a positive sign. See Figure 2.2. Note $lk(K, m_K) = 1$.

<figure><img src="image_35.png" /><figcaption>Figure 2.2: A meridian $m_K$ of a knot $K$</figcaption></figure>

**Lemma 2.6.** Let $R$ be a $\mathbb{Q}$-sphere or a $\mathbb{Q}$-ball. Let $K$ be a knot in $R$. Then $H_1(R \setminus K; \mathbb{Q})$ is equal to $\mathbb{Q}[m_K]$. Let $J$ be a knot of $R$ disjoint from $K$. Then we have $[J] = lk(J, K)[m_K]$ in $H_1(R \setminus K; \mathbb{Q})$. This equality provides an alternative definition for the linking number.
PROOF: Exercise. Note that a chain bounded by $J$ transverse to $K$ in $R$ provides a rational cobordism between $J$ and a combination of meridians of $K$. $\square$

The reader is also invited to check that the Gauss linking number $lk_G$ of Subsection 1.2.1 coincides with the above linking number $lk$ for two-component links of $S^3$, as an exercise. This is proved in the following subsection, see Proposition 2.9.

### 2.2.2 Generalizing the Gauss definition of the linking number and identifying the definitions

Let $X$ and $Y$ be two topological spaces. Recall that a homotopy from a continuous map $f$ from $X$ to $Y$ to another such $g$ is a continuous map $H: [0, 1] \times X \to Y$ such that for any $x \in X$, $H(0, x) = f(x)$ and $H(1, x) = g(x)$. Two continuous maps $f$ and $g$ from $X$ to $Y$ are said to be homotopic if there exists a homotopy from $f$ to $g$. A continuous map $f$ from $X$ to $Y$ is a homotopy equivalence if there exists a continuous map $g$ from $Y$ to $X$ such that $g \circ f$ is homotopic to the identity map of $X$ and $f \circ g$ is homotopic to the identity map of $Y$. The topological spaces $X$ and $Y$ are said to be homotopy equivalent, or of the same homotopy type if there exists a homotopy equivalence from $X$ to $Y$. Appendix A describes the homology $H_*$ of topological spaces and continuous maps.

Let $\Delta((\mathbb{R}^3)^2)$ denote the diagonal of $(\mathbb{R}^3)^2$.

**Lemma 2.7.** *The map*

$$ p_{S^2}: \begin{array}{ccc} (\mathbb{R}^3)^2 \setminus \Delta((\mathbb{R}^3)^2) & \to & S^2 \\ (x, y) & \mapsto & \frac{1}{\|y-x\|}(y-x) \end{array} $$

*is a homotopy equivalence. In particular*

$$ H_i(p_{S^2}): H_i((\mathbb{R}^3)^2 \setminus \Delta((\mathbb{R}^3)^2); \mathbb{Z}) \to H_i(S^2; \mathbb{Z}) $$

*is an isomorphism for all integer $i$, the space $(\mathbb{R}^3)^2 \setminus \Delta((\mathbb{R}^3)^2)$ is a homology $S^2$, and $[S] = H_2(p_{S^2})^{-1}[S^2]$ is a canonical generator of*

$$ H_2((\mathbb{R}^3)^2 \setminus \Delta((\mathbb{R}^3)^2); \mathbb{Z}) = \mathbb{Z}[S]. $$

PROOF: The configuration space $((\mathbb{R}^3)^2 \setminus \Delta((\mathbb{R}^3)^2))$ is homeomorphic to $\mathbb{R}^3 \times ]0, \infty[ \times S^2$ via the map

$$ (x, y) \mapsto (x, \|y - x\|, p_{S^2}(x, y)). $$
<figure><img src="image_36.png" /></figure>

As in Subsection 1.2.1, consider a two-component link $J \sqcup K: S^1 \sqcup S^1 \hookrightarrow \mathbb{R}^3$. This embedding induces an embedding

$$ J \times K: \begin{array}{ccc} S^1 \times S^1 & \hookrightarrow & (\mathbb{R}^3)^2 \setminus \Delta((\mathbb{R}^3)^2) \\ (w, z) & \mapsto & (J(w), K(z)). \end{array} $$

The map $p_{JK}$ of Subsection 1.2.1 is the composition $p_{S^2} \circ (J \times K)$. We have

$$ H_2(p_{JK}) [S^1 \times S^1] = \deg(p_{JK}) [S^2] = lk_G(J, K) [S^2] $$

in $H_2(S^2; \mathbb{Z}) = \mathbb{Z}[S^2]$. Thus we get

$$ [(J \times K)(S^1 \times S^1)] = H_2(J \times K) [S^1 \times S^1] = lk_G(J, K) [S] $$

in $H_2((\mathbb{R}^3)^2 \setminus \Delta((\mathbb{R}^3)^2); \mathbb{Z}) = \mathbb{Z}[S]$. We will see that this definition of $lk_G$ generalizes to links in rational homology spheres. Then we will prove that our generalized definition coincides with the general definition of linking numbers in this case.

For a manifold $M$, the normal bundle to the diagonal of $M^2$ in $M^2$ is identified with the tangent bundle to $M$, fiberwise, by the map

$$ [(u, v)] \in \frac{(T_x M)^2}{\Delta((T_x M)^2)} \mapsto (v - u) \in T_x M. $$

A parallelization $\tau$ of an oriented 3-manifold $M$ is a (smooth) bundle isomorphism $\tau: M \times \mathbb{R}^3 \to TM$ that restricts to $x \times \mathbb{R}^3$ as an orientation-preserving linear isomorphism from $x \times \mathbb{R}^3$ to $T_x M$, for any $x \in M$. It has long been known that any oriented 3-manifold is parallelizable (i.e., admits a parallelization). It is proved in Section 5.2. Therefore, a tubular neighborhood of the diagonal $\Delta(M^2)$ in $M^2$ is diffeomorphic to $M \times \mathbb{R}^3$.

**Lemma 2.8.** Let $R$ be a rational homology sphere and let $\infty$ be a point of $R$. Set $\tilde{R} = R \setminus \{\infty\}$. Then $\tilde{R}^2 \setminus \Delta(\tilde{R}^2)$ has the same rational homology as $S^2$. Let $B$ be a ball in $\tilde{R}$ and let $x$ be a point inside $B$, then the class $[S]$ of $x \times \partial B$ is a canonical generator of $H_2(\tilde{R}^2 \setminus \Delta(\tilde{R}^2); \mathbb{Q}) = \mathbb{Q}[S]$.

**PROOF:** In this proof, the homology coefficients are in $\mathbb{Q}$. We refer the reader to Section A.1. Since $\tilde{R}$ has the homology of a point, the Künneth Formula (Theorem A.10) implies that $\tilde{R}^2$ has the homology of a point. The excision axiom yields

$$ \begin{align*} H_*(\tilde{R}^2, \tilde{R}^2 \setminus \Delta(\tilde{R}^2)) &\cong H_*(\tilde{R} \times \mathbb{R}^3, \tilde{R} \times (\mathbb{R}^3 \setminus \{0\})) \\ &\cong H_*(\mathbb{R}^3, S^2) \cong \begin{cases} \mathbb{Q} & \text{if } * = 3, \\ 0 & \text{otherwise.} \end{cases} \end{align*} $$
Using the long exact sequence of the pair $(\tilde{R}^2, \tilde{R}^2 \setminus \Delta(\tilde{R}^2))$, we get $H_*(\tilde{R}^2 \setminus \Delta(\tilde{R}^2)) \cong H_*(S^2)$. □

Define the Gauss linking number of two disjoint links $J$ and $K$ in $\tilde{R}$ so that

$$[(J \times K)(S^1 \times S^1)] = lk_G(J, K)[S]$$

in $H_2(\tilde{R}^2 \setminus \Delta(\tilde{R}^2); \mathbb{Q})$. Note that the two definitions of $lk_G$ coincide when $\tilde{R} = \mathbb{R}^3$.

**Proposition 2.9.** For two disjoint links $J$ and $K$ in $\tilde{R}$, we have

$$lk_G(J, K) = lk(J, K)$$

**PROOF:** First recall that $lk(J, K)$ is the algebraic intersection $\langle J, \Sigma_K \rangle_R$ of $J$ and a rational chain $\Sigma_K$ bounded by $K$. Note that the definitions of $lk(J, K)$ and $lk_G(J, K)$ make sense when $J$ and $K$ are disjoint links. If $J$ has several components $J_i$, for $i = 1, \dots, n$, then $lk_G(\cup_{i=1}^n J_i, K) = \sum_{i=1}^n lk_G(J_i, K)$ and $lk(\cup_{i=1}^n J_i, K) = \sum_{i=1}^n lk(J_i, K)$. There is no loss of generality in assuming that $J$ is a knot for the proof, which we do.

The chain $\Sigma_K$ provides a rational cobordism $C$ in $\tilde{R} \setminus J$ between $K$ and a combination of meridians of $J$. Thus, it provides the rational cobordism $C \times J$ in $\tilde{R}^2 \setminus \Delta(\tilde{R}^2)$, which allows us to see that $[J \times K] = lk(J, K)[J \times m_J]$ in $H_2(\tilde{R}^2 \setminus \Delta(\tilde{R}^2); \mathbb{Q})$. Similarly, $\Sigma_J$ provides a rational cobordism between $J$ and a meridian $m_{m_J}$ of $m_J$. So $[J \times m_J] = [m_{m_J} \times m_J]$ in $H_2(\tilde{R}^2 \setminus \Delta(\tilde{R}^2); \mathbb{Q})$. This shows

$$lk_G(J, K) = lk(J, K)lk_G(m_{m_J}, m_J).$$

Thus, it remains to prove that $lk_G(m_{m_J}, m_J) = 1$ for a positive Hopf link $(m_{m_J}, m_J)$, as in Figure 1.4, in a standard ball embedded in $\tilde{R}$. Now, there is no loss of generality in assuming that our link is a Hopf link in $\mathbb{R}^3$, so the equality follows from that for the positive Hopf link in $\mathbb{R}^3$. □

**Remark 2.10.** Under the assumptions of Lemma 2.4, the reader can prove as an exercise that if $M$ is connected and if $B$ is a compact ball of $M$ containing a point $x$ in its interior, then $J \times K$ is homologous to $lk(J, K)(x \times \partial B)$ in $M^2 \setminus \Delta(M^2)$. In particular, Proposition 2.9 generalizes to all pairs $(J, K)$ as in Lemma 2.4 naturally. This justifies our sign convention in Lemma 2.4.
# Chapter 3

## Propagators

For a two-component link $(J, K)$ in $\mathbb{R}^3$, the definition of the linking number $lk(J, K)$ can be rewritten as

$$lk(J, K) = \int_{J \times K} p_{S^2}^*(\omega) = \langle J \times K, p_{S^2}^{-1}(Y) \rangle_{(\mathbb{R}^3)^2 \setminus \Delta((\mathbb{R}^3)^2)}$$

for any 2-form $\omega$ of $S^2$ such that $\int_{S^2} \omega = 1$, and for any regular value $Y$ of $p_{JK}$. Thus, $lk(J, K)$ is the integral of a 2-form $p_{S^2}^*(\omega)$ of $(\mathbb{R}^3)^2 \setminus \Delta((\mathbb{R}^3)^2)$ along the 2-cycle $[J \times K]$, or it is the intersection of the 2-cycle $[J \times K]$ with the 4-manifold $p_{S^2}^{-1}(Y)$. In order to adapt these definitions of the linking number to punctured rational homology 3-spheres $\tilde{R} = R \setminus \{\infty\}$ and to build other invariants of links and rational homology spheres $R$, we compactify $(\tilde{R})^2 \setminus \Delta((\tilde{R})^2)$ to a compact 6-manifold $C_2(R)$, in Section 3.2, using differential blow-ups described in Section 3.1. The above form $p_{S^2}^*(\omega)$ extends to $C_2(S^3 = \mathbb{R}^3 \cup \{\infty\})$ as a model propagating form. The closure of $p_{S^2}^{-1}(Y)$ in $C_2(S^3)$ is a model propagating chain. We define general propagating forms and propagating chains in $C_2(R)$ in Section 3.3 by their behaviors on the created boundary of $C_2(R)$. The linking number in a rational homology sphere $R$ is expressed in terms of these propagators as in the above equation, in Lemma 3.12. These propagators are the main ingredient in the definitions of the invariant $\mathcal{Z}$ studied in this book.

### 3.1 Blowing up in real differential topology

For a vector space $T$, $S(T)$ denotes the quotient $S(T) = (T \setminus \{0\})/\mathbb{R}^{+*}$, where $\mathbb{R}^{+*}$ acts by scalar multiplication. Recall that the unit normal bundle of a submanifold $C$ in a smooth manifold is the fiber bundle whose fiber over $x \in C$ is $SN_x(C) = S(N_x(C))$.
In this book, blowing up a submanifold $C$ in a smooth manifold $A$ is a canonical process, which transforms $A$ into a smooth manifold $\mathcal{B}(A, C)$ by replacing $C$ with the total space of its unit normal bundle. Unlike blow-ups in algebraic geometry, this blow-up of differential topology, which amounts to removing an open tubular neighborhood (thought of as infinitely small) of $C$, topologically, creates boundaries. Let us define it formally.

A smooth submanifold transverse to the ridges of a smooth manifold $A$ is a subset $C$ of $A$ such that for any point $x \in C$ there exists a smooth open embedding $\phi$ from $\mathbb{R}^c \times \mathbb{R}^e \times [0, 1]^d$ into $A$ such that $\phi(0) = x$ and the image of $\phi$ intersects $C$ exactly along $\phi(0 \times \mathbb{R}^e \times [0, 1]^d)$. Here $c$ is the codimension of $C$, $d$ and $e$ are integers, which depend on $x$.

**Definition 3.1.** Let $C$ be a smooth submanifold transverse to the ridges of a smooth manifold $A$. The blow-up $\mathcal{B}(A, C)$ is the unique smooth manifold $\mathcal{B}(A, C)$ (with possible ridges) equipped with a canonical smooth projection

$$p_b: \mathcal{B}(A, C) \to A$$

called the *blowdown map* such that

1. the restriction of $p_b$ to $p_b^{-1}(A \setminus C)$ is a canonical diffeomorphism onto $A \setminus C$, which identifies $p_b^{-1}(A \setminus C)$ with $A \setminus C$ (we will simply regard $A \setminus C$ as a subset of $\mathcal{B}(A, C)$ via this identification),

2. there is a canonical identification of $p_b^{-1}(C)$ with the total space $SN(C)$ of the unit normal bundle to $C$ in $A$,

3. the restriction of $p_b$ to $p_b^{-1}(C) = SN(C)$ is the bundle projection from $SN(C)$ to $C$,

4. any smooth diffeomorphism $\phi$ from $\mathbb{R}^c \times \mathbb{R}^e \times [0, 1]^d$ onto an open subset $\phi(\mathbb{R}^c \times \mathbb{R}^e \times [0, 1]^d)$ of $A$ whose image intersects $C$ exactly along $\phi(0 \times \mathbb{R}^e \times [0, 1]^d)$, for natural integers $c, e, d$, provides a smooth embedding

$$ \begin{array}{ccc} [0, \infty[ \times S^{c-1} \times (\mathbb{R}^e \times [0, 1]^d) & \xrightarrow{\tilde{\phi}} & \mathcal{B}(A, C) \\ (\lambda \in ]0, \infty[, v, x) & \mapsto & \phi(\lambda v, x) \\ (0, v, x) & \mapsto & T\phi(0, x)(v) \in SN(C) \end{array} $$

with open image in $\mathcal{B}(A, C)$.

**PROOF THAT THE DEFINITION IS CONSISTENT:** Use local diffeomorphisms of the form $\tilde{\phi}$ and charts on $A \setminus C$ to construct an atlas for $\mathcal{B}(A, C)$. These charts are obviously compatible over $A \setminus C$. We check compatibility for
charts $\tilde{\phi}$ and $\tilde{\psi}$ induced by embeddings $\phi$ and $\psi$ as in the statement. For those, transition maps may be written as

$$ (\lambda, u, x) \mapsto (\tilde{\lambda} = \|p_1 \circ \psi^{-1} \circ \phi(\lambda u, x)\|, \tilde{u}, \tilde{x} = p_2 \circ \psi^{-1} \circ \phi(\lambda u, x)), $$

where $p_1$ and $p_2$ respectively denote the projections on the first and the second factors of $\mathbb{R}^c \times (\mathbb{R}^e \times [0, 1]^d)$, and

$$ \tilde{u} = \begin{cases} \frac{p_1 \circ \psi^{-1} \circ \phi(\lambda u, x)}{\tilde{\lambda}} & \text{if } \lambda \neq 0 \\ \frac{T(p_1 \circ \psi^{-1} \circ \phi)(0,x)(u)}{\|T(p_1 \circ \psi^{-1} \circ \phi)(0,x)(u)\|} & \text{if } \lambda = 0 \end{cases} $$

In order to check that this is smooth, write

$$ p_1 \circ \psi^{-1} \circ \phi(\lambda u, x) = \lambda \int_0^1 T(p_1 \circ \psi^{-1} \circ \phi)(t\lambda u, x)(u)dt, $$

and check that the integral does not vanish when $\lambda$ is small enough. The restriction to $S^{c-1}$ of $T(p_1 \circ \psi^{-1} \circ \phi)(0, x)$ is an injection. So for any $u_0 \in S^{c-1}$, there exists a neighborhood of $(0, u_0)$ in $\mathbb{R} \times S^{c-1}$ such that for any $(\lambda, u)$ in this neighborhood, we have the following condition about the scalar product

$$ \langle T(p_1 \circ \psi^{-1} \circ \phi)(\lambda u, x)(u), T(p_1 \circ \psi^{-1} \circ \phi)(0, x)(u) \rangle > 0. $$

Therefore, there exists $\varepsilon > 0$ such that for any $\lambda \in ]-\varepsilon, \varepsilon[$ and for any $u \in S^{c-1}$, we have

$$ \langle T(p_1 \circ \psi^{-1} \circ \phi)(\lambda u, x)(u), T(p_1 \circ \psi^{-1} \circ \phi)(0, x)(u) \rangle > 0. $$

Then

$$ \tilde{\lambda} = \lambda \left\| \int_{0}^{1} T(p_{1} \circ \psi^{-1} \circ \phi)(t\lambda u, x)(u)dt \right\| $$

is a smooth function (defined even when $\lambda \le 0$) and

$$ \tilde{u} = \frac{\int_{0}^{1} T(p_{1} \circ \psi^{-1} \circ \phi)(t\lambda u, x)(u)dt}{\|\int_{0}^{1} T(p_{1} \circ \psi^{-1} \circ \phi)(t\lambda u, x)(u)dt\|} $$

is smooth, too. Thus, our atlas is compatible. This defines $\mathcal{B}(A, C)$ together with its smooth structure. The projection $p_b$ maps $\tilde{\phi}(\lambda, v, x)$ to $\phi(\lambda v, x)$ in a chart as above. Thus, it is obviously smooth, and it has the desired properties. $\square$

Note the following immediate proposition.
**Proposition 3.2.** The blown-up manifold $\mathcal{B}(A, C)$ is homeomorphic to the complement in $A$ of an open tubular neighborhood of $C$. In particular, $\mathcal{B}(A, C)$ is homotopy equivalent to $A \setminus C$. If $C$ and $A$ are compact, then $\mathcal{B}(A, C)$ is compact and it is a smooth compactification of $A \setminus C$.
□

Figure 3.1 first shows the result of blowing up $(0,0)$ in $\mathbb{R}^2$. The closures in $\mathcal{B}(\mathbb{R}^2, (0,0))$ of $\{0\} \times \mathbb{R}^*$, $\mathbb{R}^* \times \{0\}$, and the diagonal of $(\mathbb{R}^*)^2$ are next blown up in $\mathcal{B}(\mathbb{R}^2, (0,0))$.

<figure><img src="image_37.png" /><figcaption>Figure 3.1: A composition of blow-ups</figcaption></figure>

**Proposition 3.3.** Let $B$ and $C$ be two smooth submanifolds transverse to the ridges of a $C^\infty$ manifold $A$. Assume that $C$ is a smooth submanifold of $B$ transverse to the ridges of $B$.

1. The closure $\overline{B \setminus C}$ of $(B \setminus C)$ in $\mathcal{B}(A, C)$ is a submanifold of $\mathcal{B}(A, C)$. It intersects

$$SN(C) \subseteq \partial \mathcal{B}(A, C)$$

as the unit normal bundle $SN_B(C)$ to $C$ in $B$. It is canonically diffeomorphic to $\mathcal{B}(B, C)$.

2. The blow-up $\mathcal{B}(\mathcal{B}(A, C), \overline{B \setminus C})$ of $\mathcal{B}(A, C)$ along $\overline{B \setminus C}$ has a canonical differential structure of a manifold with ridges. The preimage of $\overline{B \setminus C} \subset \mathcal{B}(A, C)$ in $\mathcal{B}(\mathcal{B}(A, C), \overline{B \setminus C})$ under the canonical projection

$$\mathcal{B}(\mathcal{B}(A, C), \overline{B \setminus C}) \to \mathcal{B}(A, C)$$

is the pull-back via the blowdown projection from $\overline{B \setminus C}$ to $B$ of the unit normal bundle to $B$ in $A$.
PROOF:

1. Let $x \in C$. It is always possible to choose an embedding $\phi$ into $A$ as in Definition 3.1 such that $\phi(0) = x$ and $\phi(\mathbb{R}^c \times \mathbb{R}^e \times [0, 1]^d)$ intersects $C$ exactly along $\phi(0 \times \mathbb{R}^e \times [0, 1]^d)$ and $B$ exactly along $\phi(0 \times \mathbb{R}^k \times [0, 1]^d)$, with $k > e$. (First choose an embedding suitable for $B$, and then modify it to suit $C$.) Look at the induced chart $\tilde{\phi}$ of $\mathcal{B}(A, C)$ near a point of $\partial\mathcal{B}(A, C)$.

The intersection of $(B \setminus C)$ with the image of $\tilde{\phi}$ is

$$ \tilde{\phi}([0, \infty[ \times (0 \times S^{k-e-1} \subset S^{e-1}) \times \mathbb{R}^e \times [0, 1]^d). $$

Thus, the closure of $(B \setminus C)$ intersects the image of $\tilde{\phi}$ as

$$ \tilde{\phi}([0, \infty[ \times (0 \times S^{k-e-1} \subset S^{e-1}) \times \mathbb{R}^e \times [0, 1]^d). $$

2. Together with the above given charts of $\overline{B \setminus C}$, the smooth injective map

$$ \mathbb{R}^{c-k+e} \times S^{k-e-1} \to S^{c-1} \\ (u, y) \mapsto \frac{(u, y)}{\|(u, y)\|} $$

identifies $\mathbb{R}^{c-k+e}$ with the fibers of the normal bundle to $\overline{B \setminus C}$ in $\mathcal{B}(A, C)$. The blow-up process will therefore replace $\overline{B \setminus C}$ with the quotient of the corresponding $(\mathbb{R}^{c-k+e} \setminus \{0\})$-bundle by $]0, \infty[$, which is of course the pull-back under the blowdown projection $(\overline{B \setminus C} \to B)$ of the unit normal bundle to $B$ in $A$.

□

The fiber $SN_c(C)$ is oriented as the boundary of a unit ball of $N_c(C)$.

## 3.2 The configuration space $C_2(R)$

Regard $S^3$ as $\mathbb{R}^3 \cup \{\infty\}$ or as two copies of $\mathbb{R}^3$ identified along $\mathbb{R}^3 \setminus \{0\}$ by the (exceptionally orientation-reversing) diffeomorphism $x \mapsto x/\|x\|^2$.

Let $(-S_\infty^2)$ denote the unit normal bundle to $\infty$ in $S^3$, so $\mathcal{B}(S^3, \infty) = \mathbb{R}^3 \cup S_\infty^2$ and $\partial\mathcal{B}(S^3, \infty) = S_\infty^2$. There is a canonical orientation-preserving diffeomorphism $p_\infty: S_\infty^2 \to S^2$, such that $x \in S_\infty^2$ is the limit in $\mathcal{B}(S^3, \infty)$ of a sequence of points of $\mathbb{R}^3$ tending to $\infty$ along any line of $\mathbb{R}^3$ directed by $p_\infty(x) \in S^2$, in the direction of the line.
Let $\tilde{B}_{1,\infty}$ (resp. $B_{1,\infty}$) denote the complement of the closed (resp. open) ball of radius one of $\mathbb{R}^3$ in $S^3$. Let $\tilde{B}_{2,\infty}$ be the complement in $S^3$ of the closed ball $B(2)$ of radius 2 in $\mathbb{R}^3$.

Fix a rational homology sphere $R$ and a point $\infty$ of $R$. Set $\tilde{R} = R \setminus \{\infty\}$. Identify a neighborhood of $\infty$ in $R$ with $\tilde{B}_{1,\infty}$. The ball $\tilde{B}_{2,\infty}$ is a smaller neighborhood of $\infty$ in $R$ via the understood identification. Then $B_R = R \setminus \tilde{B}_{2,\infty}$ is a compact rational homology ball diffeomorphic to $\mathcal{B}(R, \infty)$.

Define the configuration space $C_2(R)$ to be the compact 6-manifold with boundary and ridges obtained from $R^2$ by blowing up first $(\infty, \infty)$ in $R^2$, and next the closures of $\{\infty\} \times \tilde{R}$, $\tilde{R} \times \{\infty\}$, and the diagonal of $\tilde{R}^2$ in $\mathcal{B}(R^2, (\infty, \infty))$, as in Figure 3.1. Then $\partial C_2(R)$ contains the unit normal bundle $((T\tilde{R}^2/\Delta(T\tilde{R}^2)) \setminus \{0\})/\mathbb{R}^{*+}$ to the diagonal of $\tilde{R}^2$. This bundle is identified with the unit tangent bundle $U\tilde{R}$ to $\tilde{R}$ by the map $([x, y] \mapsto [y - x])$.

**Lemma 3.4.** Let $\tilde{C}_2(R) = \tilde{R}^2 \setminus \Delta(\tilde{R}^2)$. The open manifold $C_2(R) \setminus \partial C_2(R)$ is $\tilde{C}_2(R)$, and the inclusion $\tilde{C}_2(R) \hookrightarrow C_2(R)$ is a homotopy equivalence. In particular, $C_2(R)$ is a compactification of $\tilde{C}_2(R)$ homotopy equivalent to $\tilde{C}_2(R)$. It has the same rational homology as the sphere $S^2$. The manifold $C_2(R)$ is a smooth compact 6-dimensional manifold with boundary and ridges. Its boundary is

$$ \partial C_2(R) = p_{R^2}^{-1}(\infty, \infty) \cup (S_{\infty}^2 \times \tilde{R}) \cup (-\tilde{R} \times S_{\infty}^2) \cup U\tilde{R}. $$

Furthermore, there is a canonical smooth projection $p_{R^2}: C_2(R) \to R^2$.

**PROOF:** This lemma is a corollary of Propositions 3.2 and 3.3, and Lemma 2.8. We give a few additional arguments to check that we can perform the three blow-ups in $\mathcal{B}(R^2, (\infty, \infty))$ simultaneously, and we take a closer look at the structure of $p_{R^2}^{-1}(\infty, \infty)$ below.

Blowing up $(\infty, \infty)$ in $B_{1,\infty}^2$ transforms a neighborhood of $(\infty, \infty)$ into the product $[0, 1[ \times S^5$. Explicitly, there is a map

$$ \begin{align*} \psi: [0, 1[ \times S^5 & \to \mathcal{B}(B_{1,\infty}^2, (\infty, \infty)) \\ (\lambda \in ]0, 1[, (x \neq 0, y \neq 0) \in S^5 \subset (\mathbb{R}^3)^2) & \mapsto \left( \frac{1}{\lambda \|x\|^2} x, \frac{1}{\lambda \|y\|^2} y \right) \\ (\lambda \in ]0, 1[, (0, y \neq 0) \in S^5 \subset (\mathbb{R}^3)^2) & \mapsto \left( \infty, \frac{1}{\lambda \|y\|^2} y \right) \\ (\lambda \in ]0, 1[, (x \neq 0, 0) \in S^5 \subset (\mathbb{R}^3)^2) & \mapsto \left( \frac{1}{\lambda \|x\|^2} x, \infty \right), \end{align*} $$

which is a diffeomorphism onto its open image.

Here, the explicit image of $(\lambda \in ]0, 1[, (x \neq 0, y \neq 0) \in S^5 \subset (\mathbb{R}^3)^2)$ is written in $(\tilde{B}_{1,\infty} \setminus \{\infty\})^2 \subset \mathcal{B}(\tilde{B}_{1,\infty}^2, (\infty, \infty))$, where $\tilde{B}_{1,\infty} \setminus \{\infty\} \subset \mathbb{R}^3$. The
image of $\psi$ is a neighborhood of the preimage of $(\infty, \infty)$ under the blowdown map

$$ \mathcal{B}\ell(R^2, (\infty, \infty)) \xrightarrow{p_1} R^2. $$

This neighborhood respectively intersects $\infty \times \check{R}$, $\check{R} \times \infty$, and $\Delta(\check{R}^2)$ as $\psi([0, 1[ \times 0 \times S^2)$, $\psi([0, 1[ \times S^2 \times 0)$, and $\psi([0, 1[ \times (S^5 \cap \Delta((\mathbb{R}^3)^2)))$. In particular, the closures of $\infty \times \check{R}$, $\check{R} \times \infty$, and $\Delta(\check{R}^2)$ in $\mathcal{B}\ell(R^2, (\infty, \infty))$ intersect the boundary $\psi(0 \times S^5)$ of $\mathcal{B}\ell(R^2, (\infty, \infty))$ as three disjoint spheres in $S^5$. They are respectively isomorphic to $\infty \times \mathcal{B}\ell(R, \infty)$, $\mathcal{B}\ell(R, \infty) \times \infty$, and $\Delta(\mathcal{B}\ell(R, \infty)^2)$. Thus, the next steps will be three blow-ups along these three disjoint smooth manifolds.

These blow-ups will preserve the product structure $\psi([0, 1[ \times .)$. Thus, $C_2(R)$ is a smooth compact 6-dimensional manifold with boundary, with three ridges $S^2 \times S^2$ in $p_{R^2}^{-1}(\infty, \infty)$. A neighborhood of these ridges in $C_2(R)$ is diffeomorphic to $[0, 1]^2 \times S^2 \times S^2$. $\square$

Let $\iota_{S^2}$ denote the antipodal map of $S^2$, which sends $x$ to $\iota_{S^2}(x) = -x$.

**Lemma 3.5.** The map $p_{S^2}$ of Lemma 2.7 extends smoothly to $C_2(S^3)$. Its extension $p_{S^2}$ satisfies

$$ p_{S^2} = \begin{cases} \iota_{S^2} \circ p_{\infty} \circ p_1 & \text{on } S_{\infty}^2 \times \mathbb{R}^3 \\ p_{\infty} \circ p_2 & \text{on } \mathbb{R}^3 \times S_{\infty}^2 \\ p_2 & \text{on } U\mathbb{R}^3 = \mathbb{R}^3 \times S^2, \end{cases} $$

where $p_1$ and $p_2$ denote the projections on the first and second factor with respect to the above expressions.

**PROOF:** Near the diagonal of $\mathbb{R}^3$, we have a chart of $C_2(S^3)$

$$ \psi_d: \mathbb{R}^3 \times [0, \infty[ \times S^2 \to C_2(S^3), $$

which maps $(x \in \mathbb{R}^3, \lambda \in ]0, \infty[, y \in S^2)$ to $(x, x + \lambda y) \in (\mathbb{R}^3)^2$. Here, $p_{S^2}$ extends as the projection onto the $S^2$ factor.

Consider the orientation-reversing embedding $\phi_{\infty}$

$$ \phi_{\infty}: \begin{array}{ccc} \mathbb{R}^3 & \to & (S^3 = \mathbb{R}^3 \cup \{\infty\}) \\ \mu(x \in S^2) & \mapsto & \begin{cases} \infty & \text{if } \mu = 0 \\ \frac{1}{\mu}x & \text{otherwise.} \end{cases} \end{array} $$

Note that this chart induces the identification $p_{\infty}$ of the unit normal bundle $S_{\infty}^2$ to $\{\infty\}$ in $S^3$ with $S^2$. When $\mu \neq 0$,

$$ p_{S^2}(\phi_{\infty}(\mu x), y \in \mathbb{R}^3) = \frac{\mu y - x}{\|\mu y - x\|}. $$
Then $p_{S^2}$ can be extended smoothly on $S_{\infty}^2 \times \mathbb{R}^3$ (where $\mu = 0$) by

$$p_{S^2}(x \in S_{\infty}^2, y \in \mathbb{R}^3) = -x.$$

Similarly, set $p_{S^2}(x \in \mathbb{R}^3, y \in S_{\infty}^2) = y$. Now, with the map $\psi$ of the proof of Lemma 3.4, if $x$ and $y$ are not equal to zero and if they are distinct, then we have

$$p_{S^2} \circ \psi((\lambda, (x, y))) = \frac{\frac{y}{\|y\|^2} - \frac{x}{\|x\|^2}}{\frac{\|y\|^2}{\|x\|^2} - \frac{x}{\|x\|^2}} = \frac{\|x\|^2 y - \|y\|^2 x}{\|x\|^2 y - \|y\|^2 x}$$

when $\lambda \neq 0$. This map extends to $\mathcal{B}(R^2, (\infty, \infty))$ outside the boundaries of $\infty \times \mathcal{B}(R, \infty)$, $\mathcal{B}(R, \infty) \times \infty$ and $\Delta(\mathcal{B}(R, \infty)^2)$, naturally, by keeping the same formula when $\lambda = 0$.

Let us check that the map $p_{S^2}$ extends smoothly over the boundary of $\Delta(\mathcal{B}(R, \infty)^2)$. There is a chart of $C_2(R)$ near the preimage of this boundary in $C_2(R)$

$$\psi_2: [0, \infty[ \times [0, \infty[ \times S^2 \times S^2 \to C_2(S^3),$$

which maps $(\lambda \in ]0, \infty[, \mu \in ]0, \infty[, x \in S^2, y \in S^2)$ to $(\phi_{\infty}(\lambda x), \phi_{\infty}(\lambda(x + \mu y)))$. With these coordinates, the map $p_{S^2}$ may be written as

$$(\lambda, \mu, x, y) \mapsto \frac{y - 2\langle x, y \rangle x - \mu x}{\|y - 2\langle x, y \rangle x - \mu x\|}.$$

Therefore, it extends smoothly along $\mu = 0$. We check that $p_{S^2}$ extends smoothly over the boundaries of $\infty \times \mathcal{B}(R, \infty)$ and $\mathcal{B}(R, \infty) \times \infty$ similarly. $\square$

<figure><img src="image_38.png" /></figure>

**Definition 3.6.** Let $\tau_s$ denote the standard parallelization of $\mathbb{R}^3$. Say that a parallelization

$$\tau: \check{R} \times \mathbb{R}^3 \to T\check{R}$$

of $\check{R}$ that coincides with $\tau_s$ on $\hat{B}_{2,\infty} \setminus \{\infty\}$ is asymptotically standard. According to Proposition 5.5, such a parallelization exists. Such a parallelization identifies $U\check{R}$ with $\check{R} \times S^2$.

**Proposition 3.7.** For any asymptotically standard parallelization $\tau$ of $\check{R}$, there exists a smooth map $p_{\tau}: \partial C_2(R) \to S^2$ such that

$$p_{\tau} = \begin{cases} p_{S^2} & \text{on } p_{R^2}^{-1}(\infty, \infty) \\ \iota_{S^2} \circ p_{\infty} \circ p_1 & \text{on } S_{\infty}^2 \times \check{R} \\ p_{\infty} \circ p_2 & \text{on } \check{R} \times S_{\infty}^2 \\ p_2 & \text{on } U\check{R} \stackrel{\tau}{=} \check{R} \times S^2, \end{cases}$$

where $p_1$ and $p_2$ denote the projections on the first and second factors with respect to the above expressions.
PROOF: This is a consequence of Lemma 3.5. □

<figure><img src="image_39.png" /></figure>

**Definition 3.8.** Define an asymptotic rational homology $\mathbb{R}^3$ to be a pair $(\tilde{R}, \tau)$, in which $\tilde{R}$ is the union over $]1, 2] \times S^2$ of a rational homology ball $B_R$ and the complement $\tilde{B}_{1,\infty} \setminus \{\infty\}$ of the unit ball in $\mathbb{R}^3$, and $\tau$ is an asymptotically standard parallelization of $\tilde{R}$. Such a pair $(\tilde{R}, \tau)$ defines the rational homology sphere $R = \tilde{R} \cup \{\infty\}$ canonically. So “Let $(\tilde{R}, \tau)$ be an asymptotic rational homology $\mathbb{R}^3$” is a shortcut for “Let $R$ be a rational homology sphere, equipped with an embedding $\phi_R: \tilde{B}_{1,\infty} \hookrightarrow R$ and an asymptotically standard parallelization $\tau$ of the complement $\tilde{R} = R \setminus \{\phi_R(\infty)\}$, with respect to $\phi_R$”. (The embedding $\phi_R$ is understood. We will not mention it anymore. We will just view it as an inclusion $\tilde{B}_{1,\infty} \hookrightarrow R$ and denote $\phi_R(\infty)$ by $\infty$.)

## 3.3 On propagators

A volume-one form of $S^2$ is a 2-form $\omega_S$ of $S^2$ such that $\int_{S^2} \omega_S = 1$. (See Appendix B for a short survey of differential forms and de Rham cohomology.)
Let $(\tilde{R}, \tau)$ be an asymptotic rational homology $\mathbb{R}^3$. Recall the map

$$p_{\tau}: \partial C_2(R) \to S^2$$

of Proposition 3.7.

**Definition 3.9.** A propagating chain of $(C_2(R), \tau)$ is a 4-chain $P$ of $C_2(R)$ such that $\partial P = p_{\tau}^{-1}(a)$ for some $a \in S^2$. A propagating form of $(C_2(R), \tau)$ is a closed 2-form $\omega$ on $C_2(R)$ whose restriction to $\partial C_2(R)$ may be expressed as $p_{\tau}^*(\omega_S)$ for some volume-one form $\omega_S$ of $S^2$. We will call propagating chains and propagating forms propagators when their nature is clear from the context.

**Examples 3.10.** Recall the map $p_{S^2}: C_2(S^3) \to S^2$ of Lemma 3.5. For any $a \in S^2$, $p_{S^2}^{-1}(a)$ is a propagating chain of $(C_2(S^3), \tau_s)$. For any 2-form $\omega_S$ of $S^2$ such that $\int_{S^2} \omega_S = 1$, $p_{S^2}^*(\omega_S)$ is a propagating form of $(C_2(S^3), \tau_s)$.

For our general $\mathbb{Q}$-sphere $R$, propagating chains of $(C_2(R), \tau)$ exist because the 3-cycle $p_{\tau}^{-1}(a)$ of $\partial C_2(R)$ bounds in $C_2(R)$ since $H_3(C_2(R); \mathbb{Q}) = 0$. Dually, propagating forms of $(C_2(R), \tau)$ exist because the restriction induces a surjective map

$$H^2(C_2(R); \mathbb{R}) \to H^2(\partial C_2(R); \mathbb{R})$$

since $H^3(C_2(R), \partial C_2(R); \mathbb{R})$ is trivial.

When $R$ is a $\mathbb{Z}$-sphere, there exist propagating chains of $(C_2(R), \tau)$ that are smooth 4-manifolds properly embedded in $C_2(R)$. See Corollary 11.10.
**Definition 3.11.** A propagating form of $C_2(R)$ is a closed 2-form $\omega_p$ on $C_2(R)$ whose restriction to $\partial C_2(R) \setminus UB_R$ is equal to $p_\tau^*(\omega)$ for some volume-one form $\omega_S$ of $S^2$ and some asymptotically standard parallelization $\tau$. Similarly, a propagating chain of $C_2(R)$ is a 4-chain $P$ of $C_2(R)$ such that $\partial P \subset \partial C_2(R)$ and $\partial P \cap (\partial C_2(R) \setminus UB_R)$ is equal to $p_\tau^{-1}(a)$ for some $a \in S^2$. (These definitions do not depend on $\tau$.)

Explicit propagating chains associated to Heegaard splittings, constructed with Greg Kuperberg, are described in Subsection 1.2.8. They are integral chains multiplied by $\frac{1}{|H_1(R; \mathbb{Z})|}$, where $|H_1(R; \mathbb{Z})|$ is the cardinality of $H_1(R; \mathbb{Z})$.

Since $C_2(R)$ is homotopy equivalent to $(\check{R}^2 \setminus \Delta (\check{R}^2))$, Lemma 2.8 ensures that $H_2(C_2(R); \mathbb{Q}) = \mathbb{Q}[S]$, where the canonical generator $[S]$ is the homology class of a fiber of $U\check{R} \subset \partial C_2(R)$. For a two-component link $(J, K)$ of $\check{R}$, the homology class $[J \times K]$ of $J \times K$ in $H_2(C_2(R); \mathbb{Q})$ is $lk(J, K)[S]$, according to Proposition 2.9.

**Lemma 3.12.** Let $(\check{R}, \tau)$ be an asymptotic rational homology $\mathbb{R}^3$. Let $C$ be a two-cycle of $C_2(R)$. For any propagating chain $P$ of $C_2(R)$ transverse to $C$ and for any propagating form $\omega$ of $C_2(R)$, we have

$$[C] = \left( \int_C \omega \right) [S] = \langle C, P \rangle_{C_2(R)} [S]$$

in $H_2(C_2(R); \mathbb{Q}) = \mathbb{Q}[S]$. In particular, we have

$$lk(J, K) = \int_{J \times K} \omega = \langle J \times K, P \rangle_{C_2(R)}.$$

for any two-component link $(J, K)$ of $\check{R}$.

**PROOF:** Fix a propagating chain $P$, the algebraic intersection $\langle C, P \rangle_{C_2(R)}$ depends only on the homology class $[C]$ of $C$ in $C_2(R)$. Similarly, since $\omega$ is closed, $\int_C \omega$ depends only on $[C]$. (Indeed, if $C$ and $C'$ cobound a chain $D$ transverse to $P$, then $C \cap P$ and $C' \cap P$ cobound $\pm(D \cap P)$, and $\int_{\partial D = C' - C} \omega = \int_D d\omega$ according to Stokes' theorem.) Furthermore, the dependence on $[C]$ is linear. Therefore, it suffices to check the lemma for a chain that represents the canonical generator $[S]$ of $H_2(C_2(R); \mathbb{Q})$. Any fiber of $U\check{R}$ is such a chain. $\square$

**Definition 3.13.** A propagating form $\omega$ of $C_2(R)$ is homogeneous if its restriction to $\partial C_2(R) \setminus UB_R$ is $p_\tau^*(\omega_{S^2})$ for the homogeneous volume form $\omega_{S^2}$ of $S^2$ of total volume 1.
**Definition 3.14.** Let $\iota$ be the involution of $C_2(R)$ that exchanges the two coordinates in $\tilde{R}^2 \setminus \Delta(\tilde{R}^2)$. An antisymmetric propagating form is a propagating form such that $\iota_*(\omega) = -\omega$.

**Example 3.15.** The propagating form $p_{S^2}^*(\omega_{S^2})$ of $(C_2(S^3), \tau)$ is antisymmetric.

The following easy lemma ensures the existence of antisymmetric propagating forms of $(C_2(R), \tau)$.

**Lemma 3.16.** Let $\omega_0$ be a propagating form of $(C_2(R), \tau)$. Then $(-\iota^*(\omega_0))$ and $\omega = \frac{1}{2}(\omega_0 - \iota^*(\omega_0))$ are propagating forms of $(C_2(R), \tau)$, and we have $\iota^*(\omega) = -\omega$. If $\omega_0$ is homogeneous, then $(-\iota^*(\omega_0))$ and $\omega = \frac{1}{2}(\omega_0 - \iota^*(\omega_0))$ are also homogeneous.

**PROOF:** Let $\omega_S$ be the volume-one form of $S^2$ such that $\omega_0|_{\partial C_2(R)} = p_\tau^*(\omega_S)$. Recall the antipodal map $\iota_{S^2}$ of $S^2$. The form $(-\iota_{S^2}^*(\omega_S))$ is a volume-one form of $S^2$, and we have $(-\iota^*(\omega_0))|_{\partial C_2(R)} = p_\tau^*(-\iota_{S^2}^*(\omega_S))$. $\square$

Also note the following lemma.

**Lemma 3.17.** Let $\omega_a$ and $\omega'_a$ be two propagating forms of $(C_2(R), \tau)$, respectively restricting to $\partial C_2(R)$ as $p_\tau^*(\omega_A)$ and $p_\tau^*(\omega'_A)$, for two volume-one forms $\omega_A$ and $\omega'_A$ of $S^2$. There exists a one-form $\eta_A$ on $S^2$ such that $\omega'_A = \omega_A + d\eta_A$. For any such $\eta_A$, there exists a one-form $\eta$ on $C_2(R)$ such that $\omega'_a - \omega_a = d\eta$ and the restriction of $\eta$ to $\partial C_2(R)$ is $p_\tau^*(\eta_A)$.

**PROOF:** Since $\omega_a$ and $\omega'_a$ are cohomologous, there exists a one-form $\eta$ on $C_2(R)$ such that $\omega'_a = \omega_a + d\eta$. Similarly, since $\int_{S^2} \omega'_A = \int_{S^2} \omega_A$, there exists a one-form $\eta_A$ on $S^2$ such that $\omega'_A = \omega_A + d\eta_A$. We have $d(\eta - p_\tau^*(\eta_A)) = 0$ on $\partial C_2(R)$. The exact sequence with real coefficients

$$0 = H^1(C_2(R)) \to H^1(\partial C_2(R)) \to H^2(C_2(R), \partial C_2(R)) \cong H_4(C_2(R)) = 0,$$
implies $H^1(\partial C_2(R); \mathbb{R}) = 0$. Therefore, there exists a function $f$ from $\partial C_2(R)$ to $\mathbb{R}$ such that

$$df = \eta - p_{\tau}^{*}(\eta_{A})$$

on $\partial C_2(R)$. Extend $f$ to a $C^\infty$ map on $C_2(R)$ and replace $\eta$ with $(\eta - df)$. $\square$
<figure><img src="image_40.png" /></figure>
# Chapter 4

## The Theta invariant

Let $(\tilde{R}, \tau)$ be an asymptotic rational homology $\mathbb{R}^3$. We are ready to define the topological invariant $\Theta$ for $(\tilde{R}, \tau)$ to be the algebraic triple intersection of three propagating chains of $(C_2(R), \tau)$, in Section 4.1. In Section 4.3, we use relative Pontrjagin classes introduced in Section 4.2 to turn $\Theta$ into a topological invariant of rational homology 3-spheres.

### 4.1 The $\Theta$-invariant of $(R, \tau)$

We defined the algebraic intersection $\langle A, B, C \rangle_D$ of three transverse compact submanifolds $A, B, C$ in a manifold $D$ such that the sum of the codimensions of $A, B$, and $C$ is the dimension of $D$ in Subsection 2.1.5. It is the sum over the intersection points $a$ of $A \cap B \cap C$ of the associated signs, where the sign of $a$ is positive if and only if the orientation of $D$ is induced by the orientation of $N_aA \oplus N_aB \oplus N_aC$, where $N_aA, N_aB$, and $N_aC$ are respectively identified with $(T_aA)^\perp, (T_aB)^\perp$, and $(T_aC)^\perp$ with the help of a Riemannian metric.

**Theorem 4.1.** Let $(\tilde{R}, \tau)$ be an asymptotic rational homology $\mathbb{R}^3$. Let $P_a, P_b$, and $P_c$ be three transverse propagators of $(C_2(R), \tau)$ with respective boundaries $p_\tau^{-1}(a), p_\tau^{-1}(b)$, and $p_\tau^{-1}(c)$ for three distinct points $a, b$, and $c$ of $S^2$. Then

$$ \Theta(R, \tau) = \langle P_a, P_b, P_c \rangle_{C_2(R)} $$

does not depend on the chosen propagators $P_a, P_b$, and $P_c$. It is a topological invariant of $(R, \tau)$. For any three propagating chains $\omega_a, \omega_b$, and $\omega_c$ of $(C_2(R), \tau)$, we have

$$ \Theta(R, \tau) = \int_{C_2(R)} \omega_a \wedge \omega_b \wedge \omega_c. $$
PROOF: Since $H_4(C_2(R); \mathbb{Q}) = 0$, if the propagator $P_a$ is replaced by a propagator $P'_a$ with the same boundary, the 4-dimensional cycle $(P'_a - P_a)$ bounds a 5-dimensional rational chain $W$ transverse to $P_b \cap P_c$. The 1-dimensional chain $W \cap P_b \cap P_c$ does not meet $\partial C_2(R)$ since $P_b \cap P_c$ does not meet $\partial C_2(R)$. Therefore, up to a well-determined sign, the boundary of $W \cap P_b \cap P_c$ is $P'_a \cap P_b \cap P_c - P_a \cap P_b \cap P_c$, as in Figure 4.1. This proves

<figure><img src="image_41.png" /><figcaption>Figure 4.1: Proof of Theorem 4.1</figcaption></figure>

that $\langle P_a, P_b, P_c \rangle_{C_2(R)}$ is independent of $P_a$ when $a$ is fixed. Similarly, it is independent of $P_b$ and $P_c$ when $b$ and $c$ are fixed. Thus, $\langle P_a, P_b, P_c \rangle_{C_2(R)}$ is a rational function on the connected set of triples $(a, b, c)$ of distinct point of $S^2$. It is easy to see that this function is continuous. Therefore, it is constant.

Let us similarly prove that $\int_{C_2(R)} \omega_a \wedge \omega_b \wedge \omega_c$ is independent of the propagating forms $\omega_a, \omega_b$, and $\omega_c$. Let $\omega'_a$ be a propagating form of $(C_2(R), \tau)$. Lemma 3.17 implies the existence of forms $\eta$ and $\eta_A$ such that $\omega'_a - \omega_a = d\eta$ and the restriction of $\eta$ to $\partial C_2(R)$ is $p_\tau^*(\eta_A)$. So we have

$$ \begin{align*} \int_{C_2(R)} \omega'_a \wedge \omega_b \wedge \omega_c - \int_{C_2(R)} \omega_a \wedge \omega_b \wedge \omega_c &= \int_{C_2(R)} d(\eta \wedge \omega_b \wedge \omega_c) \\ &= \int_{\partial C_2(R)} \eta \wedge \omega_b \wedge \omega_c \\ &= \int_{\partial C_2(R)} p_\tau^*(\eta_A \wedge \omega_B \wedge \omega_C) = 0 \end{align*} $$

since any 5-form on $S^2$ vanishes. Thus, $\int_{C_2(R)} \omega_a \wedge \omega_b \wedge \omega_c$ is independent of the propagating forms $\omega_a, \omega_b$, and $\omega_c$. Now, we can choose the propagating forms $\omega_a, \omega_b$, and $\omega_c$ supported in tiny neighborhoods of $P_a, P_b$, and $P_c$ and respectively Poincaré dual to $P_a, P_b$, and $P_c$. So the intersection of the three supports is a very small neighborhood of $P_a \cap P_b \cap P_c$, and we can easily see that $\int_{C_2(R)} \omega_a \wedge \omega_b \wedge \omega_c = \langle P_a, P_b, P_c \rangle_{C_2(R)}$. See Section 11.4, Section B.2, and Lemma B.4 in particular, for more details. $\square$

In particular, $\Theta(R, \tau)$ is equal to $\int_{C_2(R)} \omega^3$ for any propagating chain $\omega$ of $(C_2(R), \tau)$. Since such a propagating chain represents the linking number, $\Theta(R, \tau)$ can be thought of as the *cube of the linking number with respect to $\tau$*.
When $\tau$ varies continuously, $\Theta(R, \tau)$ varies continuously in $\mathbb{Q}$. So $\Theta(R, \tau)$ is an invariant of the homotopy class of $\tau$.

**Remark 4.2.** Define a *combing* of $\tilde{R}$ to be a section of $U\tilde{R}$ that coincides with $\tau_s(v)$ outside $B_R$, for some unit vector $v$ of $\mathbb{R}^3$. Let $X$ be a combing of $\tilde{R}$. Define a *propagating chain of* $(C_2(R), X)$ to be a propagating chain of $C_2(R)$ that intersects $U\tilde{R}$ along the image of $X$. Define $\tilde{\Theta}(R, X)$ to be the algebraic intersection of a propagating chain of $(C_2(R), X)$, a propagating chain of $(C_2(R), -X)$, and any other propagating chain of $C_2(R)$. It is easily proved in [Les15a, Theorem 2.1] that $\tilde{\Theta}(R, X)$ depends only on $R$ and on the homotopy class of $X$ among combings. In particular, $\Theta(R, \tau) = \tilde{\Theta}(R, \tau(v))$ depends only on the homotopy class of the combing $\tau(v)$ of $U\tilde{R}$, for some unit vector $v$ of $\mathbb{R}^3$. Further properties of the invariant $\tilde{\Theta}(R, \cdot)$ of combings are studied in [Les15c]. I found an explicit formula [Les15a, Theorem 3.8] for the invariant $\tilde{\Theta}(R, \cdot)$ from a Heegaard diagram of $R$. I computed it directly using the above definition of $\tilde{\Theta}(R, \cdot)$ together with propagators associated to Morse functions constructed with Greg Kuperberg in [Les15a].

**Example 4.3.** Using (disjoint!) propagators $p_{S^2}^{-1}(a)$, $p_{S^2}^{-1}(b)$, $p_{S^2}^{-1}(c)$ associated to three distinct points $a, b$, and $c$ of $\mathbb{R}^3$, as in Example 3.10, it is clear that

$$ \Theta(S^3, \tau_s) = \langle p_{S^2}^{-1}(a), p_{S^2}^{-1}(b), p_{S^2}^{-1}(c) \rangle_{C_2(S^3)} $$

is zero.

## 4.2 Parallelizations of 3-manifolds and Pontrjagin classes

In this subsection, $M$ denotes a smooth, compact oriented 3-manifold with possible boundary $\partial M$. Recall that a well-known theorem reproved in Section 5.2 ensures that such a 3-manifold is parallelizable.

Let $GL^+(\mathbb{R}^3)$ denote the group of orientation-preserving linear isomorphisms of $\mathbb{R}^3$. Let $[(M, \partial M), (GL^+(\mathbb{R}^3), 1)]_m$ denote the set of (continuous) maps

$$ g: (M, \partial M) \to (GL^+(\mathbb{R}^3), 1) $$

from $M$ to $GL^+(\mathbb{R}^3)$ that send $\partial M$ to the identity element 1 of $GL^+(\mathbb{R}^3)$. Let $[(M, \partial M), (GL^+(\mathbb{R}^3), 1)]$ denote the group of homotopy classes of such maps, with the group structure induced by the multiplication of maps, using the multiplication in $GL^+(\mathbb{R}^3)$. For a map $g$ in $[(M, \partial M), (GL^+(\mathbb{R}^3), 1)]_m$, define

$$ \psi_{\mathbb{R}}(g): \begin{array}{ccc} M \times \mathbb{R}^3 & \to & M \times \mathbb{R}^3 \\ (x, y) & \mapsto & (x, g(x)(y)). \end{array} $$
Let $\tau_M: M \times \mathbb{R}^3 \to TM$ be a parallelization of $M$. Then any parallelization $\tau$ of $M$ that coincides with $\tau_M$ on $\partial M$ may be written as

$$ \tau = \tau_M \circ \psi_{\mathbb{R}}(g) $$

for some $g \in [(M, \partial M), (GL^+(\mathbb{R}^3), 1)]_m$.

Thus, fixing $\tau_M$ identifies the set of homotopy classes of parallelizations of $M$ fixed on $\partial M$ with the group $[(M, \partial M), (GL^+(\mathbb{R}^3), 1)]$. Since $GL^+(\mathbb{R}^3)$ deformation retracts onto the group $SO(3)$ of orientation-preserving linear isometries of $\mathbb{R}^3$, the group $[(M, \partial M), (GL^+(\mathbb{R}^3), 1)]$ is isomorphic to $[(M, \partial M), (SO(3), 1)]$.

**Definition 4.4.** We regard $S^3$ as $B^3/\partial B^3$, where $B^3$ is the standard unit ball of $\mathbb{R}^3$ viewed as the quotient of $[0, 1] \times S^2$ where all points of $\{0\} \times S^2$ are identified with each other. Let $\chi_\pi: [0, 1] \to [0, 2\pi]$ be an increasing smooth bijection whose derivatives vanish at 0 and 1 such that $\chi_\pi(1-\theta) = 2\pi - \chi_\pi(\theta)$ for any $\theta \in [0, 1]$. Let $\rho: B^3 \to SO(3)$ be the map that sends $(\theta \in [0, 1], v \in S^2)$ to the rotation $\rho(\chi_\pi(\theta); v)$ with axis directed by $v$ and with angle $\chi_\pi(\theta)$.

This map $\rho$ induces the double covering map $\tilde{\rho}: S^3 \to SO(3)$, which orients $SO(3)$ and which allows one to deduce the first three homotopy groups of $SO(3)$ from those of $S^3$. The first three homotopy groups of $SO(3)$ are $\pi_1(SO(3)) = \mathbb{Z}/2\mathbb{Z}$, $\pi_2(SO(3)) = 0$, and $\pi_3(SO(3)) = \mathbb{Z}[\tilde{\rho}]$. For $v \in S^2$, $\pi_1(SO(3))$ is generated by the class of the loop that maps $\exp(i\theta) \in S^1$ to the rotation $\rho(\theta; v)$. See Section A.2 and Theorem A.14 in particular.

Note that a map $g$ from $(M, \partial M)$ to $(SO(3), 1)$ has a degree $\deg(g)$. The degree $\deg(g)$ is the differential degree at a regular value (different from 1) of $g$. It can also be defined homologically, by $H_3(g) [M, \partial M] = \deg(g) [SO(3), 1]$.

We prove the following theorem, for which we claim no originality, in Chapter 5, as a direct consequence of Definition 5.13, Lemmas 5.2, 5.7, 5.8, and Propositions 5.10, 5.22, and 5.26.

**Theorem 4.5.** For any compact connected oriented 3-manifold $M$, the group

$$ [(M, \partial M), (SO(3), 1)] $$

is abelian and the degree

$$ \deg: [(M, \partial M), (SO(3), 1)] \to \mathbb{Z} $$

is a group homomorphism, which induces an isomorphism

$$ \deg: [(M, \partial M), (SO(3), 1)] \otimes_{\mathbb{Z}} \mathbb{Q} \to \mathbb{Q}. $$
When $\partial M$ is equal to $\emptyset$ or $S^2$, there exists a canonical map $p_1$ from the set of homotopy classes of parallelizations of $M$—that coincide with $\tau_s$ near $S^2$ if $\partial M = S^2$—to $\mathbb{Z}$ such that

* we have $p_1((\tau_s)|_{B^3}) = 0$ and

* for any map $g$ in $[(M, \partial M), (SO(3), 1)]_m$ and for any trivialization $\tau$ of $TM$, we have

$$p_1(\tau \circ \psi_{\mathbb{R}}(g)) - p_1(\tau) = 2 \deg(g).$$

Definition 5.13 of the map $p_1$ involves relative Pontrjagin classes. When $\partial M = \emptyset$, the map $p_1$ coincides with the map $h$ studied by Friedrich Hirzebruch in [Hir73, §3.1], and by Robion Kirby and Paul Melvin in [KM99] under the name of *Hirzebruch defect*.

Since $[(M, \partial M), (SO(3), 1)]$ is abelian, the set of parallelizations of $M$ that are fixed on $\partial M$ is an affine space with translation group $[(M, \partial M), (SO(3), 1)]$.

Recall the map $\rho: B^3 \to SO(3)$ from Definition 4.4. Let $M$ be an oriented connected 3-manifold with possible boundary. For a ball $B^3$ embedded in $M$, let $\rho_M(B^3) \in [(M, \partial M), (SO(3), 1)]_m$ be a smooth map that coincides with $\rho$ on $B^3$ and that maps the complement of $B^3$ to the identity element of $SO(3)$. The homotopy class of $\rho_M(B^3)$ is well-defined.

**Lemma 4.6.** We have $\deg(\rho_M(B^3)) = 2$.

PROOF: Exercise.

□

## 4.3 Defining a $\mathbb{Q}$-sphere invariant from $\Theta$

Recall that an asymptotic rational homology $\mathbb{R}^3$ is a pair $(\tilde{R}, \tau)$, where $\tilde{R}$ is decomposed as the union over $]1, 2] \times S^2$ of a rational homology ball $B_R$ and the complement $\tilde{B}_{1,\infty} \setminus \{\infty\}$ of the unit ball of $\mathbb{R}^3$, and $\tau$ is an asymptotically standard parallelization of $\tilde{R}$. In this subsection, we prove the following proposition.

**Proposition 4.7.** Let $(\tilde{R}, \tau)$ be an asymptotic rational homology $\mathbb{R}^3$. For any map $g$ in $[(B_R, B_R \cap \tilde{B}_{1,\infty}), (SO(3), 1)]_m$ trivially extended to $\tilde{R}$, we have

$$\Theta(R, \tau \circ \psi_{\mathbb{R}}(g)) - \Theta(R, \tau) = \frac{1}{2} \deg(g).$$

Theorem 4.5 allows us to derive the following corollary from Proposition 4.7.
**Corollary 4.8.** Set $\Theta(R) = \Theta(R, \tau) - \frac{1}{4}p_1(\tau)$. Then $\Theta(R)$ is an invariant of $\mathbb{Q}$-spheres.

□

Since $p_1(\tau_s) = 0$, Example 4.3 shows that $\Theta(S^3) = 0$. We will prove $\Theta(-R) = -\Theta(R)$ for any $\mathbb{Q}$-sphere $R$ in Proposition 5.15.

More properties of $\Theta$ will appear later in this book. We will first view this invariant as the degree one part of a much more general invariant $\mathcal{Z}$ or $\mathfrak{z}$ (introduced in Theorem 7.20 and in Corollary 10.9, respectively) in Corollary 10.11. The multiplicativity of $\mathcal{Z}$ under connected sum stated in Theorem 10.26 implies that $\Theta$ is additive under connected sum. The invariant $\Theta$ will be identified with $6\lambda_{CW}$, where $\lambda_{CW}$ is the Walker generalization of the Casson invariant to $\mathbb{Q}$-spheres, in Section 18.6. See Theorem 18.31. (The Casson–Walker invariant $\lambda_{CW}$ is normalized as $\frac{1}{2}\lambda_W$ for rational homology spheres, where $\lambda_W$ is the Walker normalisation in [Wal92].) The equality $\Theta = 6\lambda_{CW}$ will be obtained as a consequence of a universality property of $\mathcal{Z}$ with respect to a theory of finite type invariants. We will also present a direct proof of a surgery formula satisfied by $\Theta$ in Section 18.3.

Let us begin the proof of Proposition 4.7 with the following lemma.

**Lemma 4.9.** The variation $\Theta(R, \tau \circ \psi_{\mathbb{R}}(g)) - \Theta(R, \tau)$ in Proposition 4.7 is independent of $\tau$. Set $\Theta'(g) = \Theta(R, \tau \circ \psi_{\mathbb{R}}(g)) - \Theta(R, \tau)$. Then $\Theta'$ is a homomorphism from $[(B_R, B_R \cap \mathring{B}_{1,\infty}), (SO(3), 1)]$ to $\mathbb{Q}$.

**PROOF:** For $d = a, b$ or $c$, the propagator $P_d$ of $(C_2(R), \tau)$ of Theorem 4.1 can be assumed to be a product $[-1, 0] \times p_\tau|_{UB_R}^{-1}(d)$ on a collar $[-1, 0] \times UB_R$ of $UB_R$ in $C_2(R)$. Since $H_3([-1, 0] \times UB_R; \mathbb{Q}) = 0$, the cycle

$$ (\partial([-1, 0] \times p_{\tau}|_{UB_R}^{-1}(d)) \setminus (\{0\} \times p_{\tau}|_{UB_R}^{-1}(d))) \cup (\{0\} \times p_{\tau \circ \psi_{\mathbb{R}}(g)}|_{UB_R}^{-1}(d)) $$

bounds a chain $G_d$.

The chains $G_a, G_b$, and $G_c$ can be assumed to be transverse. Construct the propagator $P_d(g)$ of $(C_2(R), \tau \circ \psi_{\mathbb{R}}(g))$ from $P_d$ by replacing $[-1, 0] \times p_\tau|_{UB_R}^{-1}(d)$ with $G_d$ on $[-1, 0] \times UB_R$. Then we have

$$ \Theta(R, \tau \circ \psi_{\mathbb{R}}(g)) - \Theta(R, \tau) = \langle G_a, G_b, G_c \rangle_{[-1,0] \times UB_R}. $$

Using $\tau$ to identify $UB_R$ with $B_R \times S^2$, the cycle $\partial G_d$ may be written as

$$ \partial G_d = ((\partial([-1, 0] \times B_R) \setminus (\{0\} \times B_R)) \times \{d\}) \cup (\{0\} \times (\cup_{m \in B_R} (m, g(m)(d)))) $$
Therefore, $\Theta(R, \tau \circ \psi_R(g)) - \Theta(R, \tau)$ is independent of $\tau$. Then it is easy to see that $\Theta'$ is a homomorphism from $[(B_R, \partial B_R), (SO(3), 1)]$ to $\mathbb{Q}$. $\square$

So, Theorem 4.5 and Lemma 4.6 reduce the proof of Proposition 4.7 to the proof that $\Theta'(\rho_R(B^3)) = 1$. It is easy to see that $\Theta'(\rho_R(B^3)) = \Theta'(\rho)$. Thus, we have reduced the proof of Proposition 4.7 to the proof of the following lemma.

**Lemma 4.10.** We have $\Theta'(\rho) = 1$.

To prove this lemma, we will compute the expression $\langle G_a, G_{-a}, G_c \rangle_{[-1,0]\times UB^3}$ from the above proof of Lemma 4.9 with explicit $G_a$ and $G_{-a}$ in $[-1,0]\times UB^3$. We will construct the chain $G_a$ from a chain $G(a)$ of $B^3 \times S^2$ described in Lemma 4.12.

Again, we regard $B^3$ as the quotient of $[0,1] \times S^2$ where all points of $\{0\} \times S^2$ are identified with each other. We first prove the following lemma:

**Lemma 4.11.** Recall the map $\rho: B^3 \to SO(3)$ from Definition 4.4. Let $a \in S^2$. The point $(-a)$ is a regular value of the map

$$ \rho_a: \begin{array}{ccc} B^3 & \to & S^2 \\ m & \mapsto & \rho(m)(a) \end{array} $$

and its preimage (cooriented by $S^2$ via $\rho_a$) is the knot $L_a = -\{\frac{1}{2}\} \times S_a$, where $S_a$ is the circle of $S^2$ of vectors orthogonal to $a$, oriented as the boundary of the hemisphere that contains $a$.

**PROOF:** We prove the lemma when $a$ is the North Pole $\vec{N}$. It is easy to see that $\rho_{\vec{N}}^{-1}(-\vec{N}) = L_{\vec{N}}$ up to orientation.

<figure><img src="image_42.png" /></figure>

Let $(\frac{1}{2}, x) \in L_{\vec{N}}$. Let $C$ denote the great circle of $S^2$ that contains $\vec{N}$ and $x$. Orient $C$ from $x$ towards $(-\vec{N})$. When $m$ moves along $\{\frac{1}{2}\} \times C$ from $(\frac{1}{2}, x)$ towards $(\frac{1}{2}, -\vec{N})$, $\rho(m)(\vec{N})$ moves from $(-\vec{N})$ along $C$ following the orientation of $C$, too. Let $v_1$ denote the oriented unit tangent vector to $C$ at $(-\vec{N})$. In our figure, $x$ is on the left, $C$ is oriented counterclockwise, and $v_1$ points to the right. So $S^2$ is oriented at $(-\vec{N})$ by $v_1$ and by the tangent vector $v_2$ at $(-\vec{N})$ towards us. In order to move $\rho(\theta; v)(\vec{N})$ in the $v_2$ direction, one increases $\theta$, so $L_{\vec{N}}$ is cooriented and oriented as in the figure. $\square$
**Lemma 4.12.** Let $a \in S^2$. Let $m \in B^3$. Recall the notation from Lemma 4.11 above where $\rho(m)(a) = \rho_a(m)$. When $m \notin L_a$, let $[a, \rho_a(m)]$ denote the unique geodesic arc of $S^2$ with length ($\ell \in [0, \pi]$) from $a$ to $\rho_a(m)$. For $t \in [0, 1]$, let $X_t(m) \in [a, \rho_a(m)]$ be such that the length of $[a, X_t(m)]$ is $t\ell$. Let $G_h(a)$ be the closure of

$$ (\cup_{t \in [0,1], m \in (B^3 \setminus L_a)} (m, X_t(m))) $$

in $B^3 \times S^2$. Let $D_a$ be the disk image of $[0, \frac{1}{2}] \times (-S_a)$ bounded by $L_a$ in $B^3$. Set

$$ G(a) = G_h(a) + D_a \times S^2. $$

Then $G(a)$ is a chain of $B^3 \times S^2$ such that

$$ \partial G(a) = -(B^3 \times a) + \cup_{m \in B^3} (m, \rho_a(m)). $$

**PROOF:** The map $X_t$ is well-defined on $(B^3 \setminus L_a)$. We have $X_0(m) = a$ and $X_1(m) = \rho_a(m)$. Let us show how the definition of $X_t$ extends smoothly to the manifold $\mathcal{B}(B^3, L_a)$ obtained from $B^3$ by blowing up $L_a$. The map $\rho_a$ maps the normal bundle to $L_a$ to a disk of $S^2$ around $(-a)$ by an orientation-preserving diffeomorphism on every fiber (near the origin). In particular, $\rho_a$ induces a map from the unit normal bundle $SN(L_a)$ to $L_a$ in $R$ to the unit normal bundle to $(-a)$ in $S^2$. This map preserves the orientation of the fibers. Then for an element $y$ of $SN(L_a)$, define $X_t(y)$ as before on the half great circle $[a, -a]_{\rho_a(-y)}$ from $a$ to $(-a)$ that is tangent to $\rho_a(-y)$ at $(-a)$. (So $\rho_a(-y)$ is an outward normal to $[a, -a]_{\rho_a(-y)}$ at $(-a)$.) This continuously extends the definition of $X_t$. So we have

$$ G_h(a) = \cup_{t \in [0,1], m \in \mathcal{B}(B^3, L_a)} (p_{B^3}(m), X_t(m)), $$

and

$$ \partial G_h(a) = -(B^3 \times a) + \cup_{m \in B^3} (m, \rho_a(m)) \\ + \cup_{t \in [0,1], m \in -\partial \mathcal{B}(B^3, L_a)} (p_{B^3}(m), X_t(m)), $$

where $(-\partial \mathcal{B}(S^3, L_a))$ is oriented as the boundary of a tubular neighborhood of $L_a$. For any $x \in L_a$, the sphere $S^2$ is covered with degree $(-1)$ by the image of $[0, 1] \times SN_x(L_a)$, where the fiber $SN_x(L_a)$ of $SN(L_a)$ is oriented as the boundary of a disk in the fiber $N_x(L_a)$ of the normal bundle. We may therefore write the last summand as $(-L_a \times S^2)$. $\square$

**PROOF OF LEMMA 4.10:** We use the notation of the proof of Lemma 4.9 and construct an explicit $G_a$ in $[-1, 0] \times UB^3 \stackrel{\tau_s}{=} [-1, 0] \times B^3 \times S^2$. Let $\iota$ be
the endomorphism of $UB^3$ over $B^3$ that maps a unit vector to the opposite one. Recall the chain $G(a)$ of Lemma 4.12. Set

$$ G_a = ([-1, -2/3] \times B^3 \times a) + (\{-2/3\} \times G(a)) \\ + ([-2/3, 0] \times \cup_{m \in B^3} (m, \rho_a(m))) $$

and

$$ G_{-a} = ([-1, -1/3] \times B^3 \times (-a)) + (\{-1/3\} \times \iota(G(a))) \\ + ([-1/3, 0] \times \cup_{m \in B^3} (m, \rho(m)(-a))). $$

Then we have

$$ G_a \cap G_{-a} = ([-2/3, -1/3] \times L_a \times (-a)) + (\{-2/3\} \times D_a \times (-a)) \\ - (\{-1/3\} \times \cup_{m \in D_a} (m, \rho_a(m))). $$

Finally, according to the proof of Lemma 4.9, $\Theta'(\rho)$ is the algebraic intersection of $G_a \cap G_{-a}$ with $P_c(\rho)$ in $C_2(R)$. This intersection coincides with the algebraic intersection of $G_a \cap G_{-a}$ with any propagator of $C_2(R)$, according to Lemma 3.12. Therefore, we have

$$ \Theta'(\rho) = \langle P_c, G_a \cap G_{-a} \rangle_{[-1,0] \times B^3 \times S^2} = - \deg_c(\rho_a : D_a \to S^2) $$

for any regular value $c \neq -a$ of $\rho_a|_{D_a}$. Since the image of the quotient $D_a$ of $[0, \frac{1}{2}] \times (-S_a)$ under $\rho_a$ covers the sphere with degree $(-1)$, we get $\Theta'(\rho) = 1$. $\square$

<figure><img src="image_43.png" /></figure>

So, Proposition 4.7 is proved.
<figure><img src="image_44.png" /></figure>
# Chapter 5

## Parallelizations of 3-manifolds and Pontrjagin classes

In this chapter, we fix a smooth oriented connected 3-manifold $M$ with possible boundary, and we study its parallelizations. In particular, we prove Theorem 4.5 in Sections 5.1, 5.3, and 5.7. We will use this theorem in our general constructions of link invariants in 3-manifolds as in the definition of $\Theta$ in Section 4.3. This chapter also describes other properties of Pontrjagin classes, which will be used in the fourth part of this book in some universality proofs. Section 5.9 describes the structure of the space of parallelizations of oriented 3-manifolds, more precisely. It is not used in other parts of the book.

### 5.1 [(M, ∂M), (SO(3), 1)] is an abelian group.

Again, we regard $S^3$ as $B^3/\partial B^3$, and $B^3$ as the quotient of $[0, 1] \times S^2$ where all points of $\{0\} \times S^2$ are identified with each other. Recall the map $\rho: B^3 \to SO(3)$ of Definition 4.4, which maps $(\theta \in [0, 1], v \in S^2)$ to the rotation $\rho(\chi_\pi(\theta); v)$ with axis directed by $v$ and with angle $\chi_\pi(\theta)$. Also recall that the group structure of $[(M, \partial M), (SO(3), 1)]$ is induced by the multiplication of maps, using the multiplication of $SO(3)$.

Any $g \in [(M, \partial M), (SO(3), 1)]_m$ induces a map

$$H_1(g; \mathbb{Z}): H_1(M, \partial M; \mathbb{Z}) \to (H_1(SO(3), 1) = \mathbb{Z}/2\mathbb{Z}).$$

Since

$$ \begin{aligned} H_1(M, \partial M; \mathbb{Z}/2\mathbb{Z}) &= H_1(M, \partial M; \mathbb{Z})/2H_1(M, \partial M; \mathbb{Z}) \\ &= H_1(M, \partial M; \mathbb{Z}) \otimes_{\mathbb{Z}} \mathbb{Z}/2\mathbb{Z}, \end{aligned} $$
we have

$$ \begin{aligned} \operatorname{Hom}(H_1(M, \partial M; \mathbb{Z}), \mathbb{Z}/2\mathbb{Z}) &= \operatorname{Hom}(H_1(M, \partial M; \mathbb{Z}/2\mathbb{Z}), \mathbb{Z}/2\mathbb{Z}) \\ &= H^1(M, \partial M; \mathbb{Z}/2\mathbb{Z}), \end{aligned} $$

and the image of $H_1(g; \mathbb{Z})$ in $H^1(M, \partial M; \mathbb{Z}/2\mathbb{Z})$ under the above isomorphisms is denoted by $H^1(g; \mathbb{Z}/2\mathbb{Z})$. (Equivalently, $H^1(g; \mathbb{Z}/2\mathbb{Z})$ denotes the image of the generator of $H^1(SO(3), 1; \mathbb{Z}/2\mathbb{Z}) = \mathbb{Z}/2\mathbb{Z}$ under $H^1(g; \mathbb{Z}/2\mathbb{Z})$ in $H^1(M, \partial M; \mathbb{Z}/2\mathbb{Z})$.)

For $v \in S^2$, let $\gamma_v$ denote the loop that maps $\exp(i\theta) \in S^1$ to the rotation $\rho(\theta; v)$ with axis directed by $v$ and with angle $\theta$. Let $\mathbb{R}P_S^2$ denote the projective plane embedded in $SO(3)$ consisting of the rotations of $SO(3)$ of angle $\pi$. Note that $\gamma_v(S^1) = \rho([0, 1] \times \{v\})$ and $\mathbb{R}P_S^2 = \rho(\{1/2\} \times S^2)$ intersect once transversally. Thus, we have $H_1(SO(3); \mathbb{Z}) = \mathbb{Z}/2\mathbb{Z}[\gamma_v]$ and $H_2(SO(3); \mathbb{Z}/2\mathbb{Z}) = \mathbb{Z}/2\mathbb{Z}[\mathbb{R}P_S^2]$.

**Lemma 5.1.** *The map*

$$ H^1(\cdot; \mathbb{Z}/2\mathbb{Z}): \begin{array}{ccc} [(M, \partial M), (SO(3), 1)] & \to & H^1(M, \partial M; \mathbb{Z}/2\mathbb{Z}) \\ [g] & \mapsto & H^1(g; \mathbb{Z}/2\mathbb{Z}) \end{array} $$

*is a group homomorphism.*

**PROOF:** Let $f$ and $g$ be two elements of $[(M, \partial M), (SO(3), 1)]_m$. In order to prove $H^1(fg; \mathbb{Z}/2\mathbb{Z}) = H^1(f; \mathbb{Z}/2\mathbb{Z}) + H^1(g; \mathbb{Z}/2\mathbb{Z})$, it suffices to prove that, for any path $\gamma$: $[0, 1] \to M$ whose image $\gamma(\partial[0, 1])$ is in $\partial M$, we have

$$ H_1(fg; \mathbb{Z}/2\mathbb{Z}) ([\gamma]) = H_1(f; \mathbb{Z}/2\mathbb{Z}) ([\gamma]) + H_1(g; \mathbb{Z}/2\mathbb{Z}) ([\gamma]), $$

where $H_1(f; \mathbb{Z}/2\mathbb{Z}) ([\gamma]) = \langle f \circ \gamma, \mathbb{R}P_S^2 \rangle_{SO(3)} [\gamma_v]$ for the mod 2 algebraic intersection $\langle f \circ \gamma, \mathbb{R}P_S^2 \rangle_{SO(3)}$. Both sides of this equation depend only on the homotopy classes of $f \circ \gamma$ and $g \circ \gamma$. Therefore, we may assume that the supports (closures of the preimages of $SO(3) \setminus \{1\}$) of $f \circ \gamma$ and $g \circ \gamma$ are disjoint and easily conclude. $\square$

**Lemma 5.2.** Let $M$ be an oriented connected 3-manifold with possible boundary. Recall that $\rho_M(B^3) \in [(M, \partial M), (SO(3), 1)]_m$ is a map that coincides with the map $\rho$ of Definition 4.4 on a ball $B^3$ embedded in $M$ and that maps the complement of $B^3$ to the unit of $SO(3)$.

1. Any homotopy class of a map $g$ from $(M, \partial M)$ to $(SO(3), 1)$, such that $H^1(g; \mathbb{Z}/2\mathbb{Z})$ is trivial, belongs to the subgroup

$$ \langle [\rho_M(B^3)] \rangle $$

of $[(M, \partial M), (SO(3), 1)]$ generated by $[\rho_M(B^3)]$.