An extension of a theorem by Cimasoni and Conway

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Dipartimento di Matematica

Corso di Laurea Magistrale in Matematica

Tesi di Laurea Magistrale

An extension of a theorem by

Cimasoni and Conway

Candidata:

Relatore:

Alice Merz

Prof. Paolo Lisca

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Introduction

This thesis is principally focused on the study of certain link isotopy invariants. By link we mean the embedding of m copies of S1 in S3.

We will focus on a generalization of the Levine-Tristram signatures in the ver-sion presented by Cimasoni and Florens in their article [3] where they are called multivariate Levine-Tristram signatures.

Levine [11] and Tristram [14] introduced a set of invariants for a link which assign to each ω ∈ S1_{\ {1} an integer in Z. These are the so-called Levine-Tristram}

signatures: for every link L they are defined starting from a Seifert surface for the link and they are indicated by σω(L).

These invariants are very important for multiple reasons: for example for ω in a dense subset of S1 they are concordance invariants and they give a lower bound on the four-genus of a link.

For every ω ∈ S1 _{\ {1} we can precompose the map L 7→ σ}

ω(L) with the map

assigning to each braid α its closure α. In this way we obtain a function from the_b n-braid group Bn to Z.

Figure 1: A 3-braid and its closure.

An immediate question is whether this map is a homomorphism for any ω ∈ S1_,

but it is straightforward to see that the only link invariant for which this property holds is the trivial one. It is however interesting to calculate how far this map is

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INTRODUCTION iv from being a homomorphism and therefore to calculate the additivity defect

σω( cαβ) − σω(α) − σb ω( bβ) for each pair α, β of braids in Bn.

Gambaudo and Ghys [9] were able to calculate this quantity in terms of a classical object in low dimensional topology: the reduced Burau representation

Bt: Bn7→ GLn−1(Z[t±1])

that we know is unitary with respect to a certain skew-Hermitian form on (Z[t±1])n−1. More precisely, their result is:

Theorem (Gambaudo and Ghys). For any α, β ∈ Bn

σω( cαβ) − σω(α) − σb ω( bβ) = − Meyer(Bω(α),Bω(β)) for every ω ∈ S1_{\ {1} of order coprime to n.}

The operator Meyer indicates the Meyer cocycle, which can be evaluated on two unitary matrices and takes values in Z.

This result is important for two main reasons:

• it relates two very well studied and important objects in knot theory, such as the Levine-Tristram signature and the reduced Burau representation;

• it gives an efficient algorithm to calculate the Levine-Tristram signature for every link L.

In [2] Cimasoni and A. Conway generalized the Gambaudo-Ghys result to the case of the multivariate Levine-Tristram signature.

The multivariate Levine-Tristram signature for a µ-coloured link (that is to say a link with each connected component coloured by an element of {1, . . . , µ}) is a function

T_∗µ _{→ Z} ω 7→ σω(L),

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INTRODUCTION v The multivariate signature is defined using a ”generalized Seifert surface” for a coloured link, called a C-complex. For µ = 1 it coincides with the Levine-Tristram signature.

Figure 2: A C-complex for a 2-coloured link.

The result by Cimasoni and Conway generalizes the formula of Gambaudo and Ghys for each ω in a certain dense subset of S1 _{in two ways:}

• it calculates the additivity defect of the multivariate signature for coloured braids;

• it extends the result to coloured tangles.

A µ-coloured tangle induces a colouring and an orientation (hence a sign) on its boundary. If we denote by c the colouring on the boundary of τ contained in D2×{0} and by c0 the colouring on the boundary of τ contained in D2_{× {1} we say that τ is}

a (c, c0)-tangle. They obtained:

Theorem (Cimasoni and Conway). For any (c, c)-tangles τ1, τ2

σω(τd1τ2) − σω(τb1) − σω(τb2) = Maslov(Fω(τ1), ∆,Fω(τ2)) for all ω = (ω1, . . . , ωµ) ∈ T∗µ such that

• each ωj has order kj ∈ N greater than one and the orders are pairwise

coprime;

• the integer ij is not 0 and is coprime to kj for every j. Here ij counts with

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INTRODUCTION vi Here τ denotes the reflection of τ along a horizontal plane, with inverted orien-tation. A restriction they encountered on extending this formula to a wider subset of S1 _{is given by the 4-dimensional interpretation of the Levine-Tristram signature}

developed in [3] by Cimasoni and Florens, which makes use of branched covers of the 4-ball. For this reason their generalization can apply only to ω’s of finite order. There is a 4-dimensional interpretation of the multivariate Levine-Tristram signature as the signature of a twisted intersection form on a certain 4-manifold associated to the link L. This interpretation was studied by Conway, Nagel and Toffoli in [6]. This thesis is focused on extending the result of Cimasoni and Conway to a wider subset of T_∗µ using this more algebraic interpretation of the Levine-Tristram signature. We start by introducing in Chapter 1 the ordinary Levine-Tristram signature and the formula of Gambaudo and Ghys.

In Chapter 2 we introduce the multivariate signature and discuss some of its basic properties.

In Chapter 3 we discuss twisted homology and cohomology, we introduce the Poincar´e duality isomorphisms and the evaluation map, which allow us to define twisted intersection form. We then move on to discuss some fundamental properties of this homological theory, such as the long exact sequence of the pair, excision and the Mayer-Vietoris sequence.

In Chapter 4 we will see an adaptation to our context of an important result of Wall [16] on the non additivity of the signature of 4k-manifolds.

In Chapter 5 we define a functor, called the isotropic functor, which generalizes in some sense the Burau representation. In fact (coloured) tangles do not form a group but they can be interpreted as the morphisms of a certain category Tangles_µ, and our isotropic functor has source in this category and target in a certain category of complex vector spaces.

Finally, in Chapter 6 we state and prove the extension of the theorem by Cimasoni and Conway, and we see that the result is indeed a generalization of the Gambaudo-Ghys formula when restricted to coloured braids.

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Chapter 1 Levine-Tristram signature and the

formula of Gambaudo and Ghys

In this chapter we introduce the Levine-Tristram signatures as studied by Levine [11] and Tristram [14] and we state the Gambaudo-Ghys formula. We start by recalling some basic facts about links.

A link is a closed 1-dimensional submanifold of S3. Being closed, it consists of finitely many S1 _{embedded in S}3_{. A connected link is called a knot. Links are usually}

considered up to ambient isotopy.

Given an oriented knot K, let ν(K) indicate a closed tubular neighbourhood of K, which is a solid torus, and let X(K) indicate the exterior of the knot S3_{\ν(K). A}

meridian of ν(K) generates H1(X(K); Z), therefore we can fix a standard generator

by choosing the meridian m oriented as in the following figure:

This gives a canonical identification H1(X(K); Z) ∼= Z.

Given two disjoint oriented knots K1 and K2, the homology class of K2 in

H1(X(K1); Z) is uniquely identified by a certain α ∈ Z. We call α the linking

number of K1 and K2 and we denote it by

lk(K1, K2).

A link diagram is an immersion of a link in R2_{, which is one-to-one except at}

some transverse double points, called crossings. At every crossing we distinguish the 1

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CHAPTER 1. L.-T. SIGNATURE AND G.-G. FORMULA 2 over-strand from the under-strand by creating a break in the strand that is passing underneath. We can assign to every intersection in a diagram a sign as in the figure:

The linking number of two knots K1and K2can be computed from a link diagram

of K1 and K2 by summing the signs of their mutual intersections and dividing by 2.

Therefore the linking number is symmetric.

Definition 1.0.1. Given an oriented link L a Seifert surface is a compact, connected, oriented surface F , embedded in S3_{, which has L as oriented boundary.}

L

Figure 1.1: A Seifert surface for the link L.

Since F is orientable, it admits a regular neighbourhood in S3 _{homeomorphic to}

F × [−1, 1] in which F corresponds to F × {0}. For ε = ±1 there are push-off maps iε : H1(F ; Z) → H1(S3\ F ; Z)

defined by sending the homology class of a curve γ to the homology class of γ × {ε}. There is a pairing, called the Seifert form:

H1(F ; Z) × H1(F ; Z) → Z

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CHAPTER 1. L.-T. SIGNATURE AND G.-G. FORMULA 3 A matrix A for this Seifert form is called a Seifert matrix. For every ω ∈ S1 _{⊂ C}

observe that the matrix

(1 − ω)A + (1 − ω)AT is Hermitian.

Definition 1.0.2. Let L be an oriented link, let F be a Seifert surface for L and let A be a matrix representing the Seifert pairing of F . Given ω ∈ S1_{, the Levine-Tristram}

signature of L at ω is defined as the signature of (1 − ω)A + (1 − ω)AT.

It can be proved that these signatures are well-defined link isotopy invariants, that is to say they are independent of the chosen Seifert surface and of the matrix representation of the Seifert form and that they give rise to a function

σL: S1\ {1} → Z

which is piecewise constant.

The Levine-Tristram signatures are important link invariants which are still well studied in low-dimensional topology. For example it is known that σL(ω) is a

concor-dance invariant for any ω root of unity of prime power order, but a precise character-ization of the subset of S1 _{for which the Levine-Tristram signatures are concordance}

invariant was given only recently by Nagel and Powell [13].

1.1 The four-dimensional interpretation

Let F denote a properly embedded, connected, oriented surface with non-empty boundary in the four-ball D4. Call WF the complement of an open tubular

neigh-bourhood of F in D4_{. It is straightforward to see that H}

1(WF) ' Z and it is generated

by a meridian of the surface. For k ∈ N there is a surjective map φ : π1(WF) → H1(WF) → Z/kZ

defined by composing the Hurewicz map with the quotient Z → Z/kZ. Let Wk→ WF

denote the cover of WF associated to ker φ. Notice that the restriction of this cover

to F × S1 ⊂ ∂WF is F × S1 id ×π

k

−−−→ F × S1_{, where π}

k is the k-fold cover of the circle.

The space WF = Wk∪F ×S1 (F × D2) gives, after extending in the obvious way the

covering map to F × D2 _{a branched cover of D}4

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CHAPTER 1. L.-T. SIGNATURE AND G.-G. FORMULA 4 branched along F = F × {0}.

More precisely, the group Z/kZ acts on the cover Wkand we extend this action to

the whole branched cover by setting it as l(p, x) = (p, xl_{) on F × D}2_{, where l ∈ Z/kZ,}

p ∈ F , x ∈ D2_{. Therefore there is a C[Z/kZ]-module structure on H}

2(WF; C). Let

α be a generator of Z/kZ and for every root of unity ω of order k define H2(WF; C)ω = {x ∈ H2(WF; C) ; α · x = ωx}.

Restricting the Hermitian intersection form on H2(WF; C)1to H2(WF; C)ωproduces

a Hermitian form, and we call its signature the ω-signature of WF, and we write

sign_ω(WF). The next result, due to Viro [15] and Kauffman-Taylor [10], gives us a

4-dimensional interpretation of the Levine-Tristram signature.

Theorem 1.1.1 ([15, 10]). Let L be an oriented link, and let F be a connected, oriented, properly embedded surface in D4 such that ∂F = L. Let WF be the

k-fold covering of D4 _{branched along F . Then, for every root of unity of order k, the}

following equality holds:

σL(ω) = signω(WF).

1.2 The Gambaudo-Ghys formula

Given a positive integer n, let p(n)_j be the point 2j−n−1_n , 0 ∈ D2 _{for j = 1, . . . , n. A}

braid with n strands is an oriented, compact, proper, one-dimensional submanifold of D2_{× [0, 1] whose oriented boundary is}

n G j=0 pj × {0} t − n G j=0 pj × {1} ! ,

with no closed connected components (therefore with just n connected components) and where the projection to [0, 1] maps each component homeomorphically onto [0, 1]. We normally consider braids up to ambient isotopy which fixes the boundary of D2× [0, 1].

Definition 1.2.1. The braid group Bn is the set of isotopy classes of braids with

n-strands. Given two braids α, β, the composition αβ is obtained by gluing α on top of β and shrinking the height of the resulting 1-manifold by a factor two. The identity is the trivial braid idn = {p

(n) 1 , . . . , p

(n)

n } × [0, 1], and it can be seen that for

every braid there is an inverse, obtained by reflecting the braid along a horizontal plane.

1_{it is the only Hermitian extension of the real intersection form to H}

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CHAPTER 1. L.-T. SIGNATURE AND G.-G. FORMULA 5

Figure 1.2: The composition of braids

+ =

Figure 1.3: The inverse of a braid

Given an n-braid α, there is a well-defined closure map, that gives us a link α:_b

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CHAPTER 1. L.-T. SIGNATURE AND G.-G. FORMULA 6 Consider an arbitrary link invariant I , which takes values in an abelian group. We can think of precomposing this invariant with the braid closure, defining a map

α 7→I (bα)

between two groups. A natural question to ask is whether this map can be a homo-morphism. However it is easy to see that the only invariant with this property is the trivial one. We can therefore try to evaluate the homomorphism defect

I (cαβ) −I (α) −_b I (bβ).

The result of Gambaudo and Ghys provides an evaluation of this defect for the Levine-Tristram signature. Before stating their result we need to introduce a funda-mental object in low dimensional topology, called the reduced Burau representation.

1.2.1 The reduced Burau representation

The Gambaudo-Ghys formula expresses the homomorphism defect of the Levine-Tristram signature in terms of the reduced Burau representation. This is a represen-tation of the braid group and is an important object of study in itself.

Recall that the braid group Bn is actually isomorphic to the mapping class group

of the disk with n-punctures Dn (see for example [8], Chapter 9). Let β ∈ Bn and

let hβ denote the automorphism of Dn related to β.

Let z ∈ Dn be a point in the boundary of the disk, and let xi be a loop with

basepoint z and winding one time counterclockwise around the i-th puncture p(n)_i . The xi’s are a basis of π1(Dn). Consider the map

ψ : π1(Dn) → Z = hti

which sends each xi to the generator t. Let π : D∞n → Dn be the regular cover

corresponding to the normal subgroup ker ψ of π1(Dn). The group Z = hti acts on

the cover D∞_n and the homology groups of D∞_n are naturally Λ-modules, where Λ denotes Z[t±1].

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CHAPTER 1. L.-T. SIGNATURE AND G.-G. FORMULA 7

Lemma 1.2.2. The Z[t±1]-module H1(Dn∞) is free of rank n − 1. If ez is a lift of z and x_ei is the lift of xi starting at z, then ve i := [exi+1] − [xei] for i = 1, . . . , n − 1 is a basis of H1(D∞n)

Proof. Since Dn is homotopy equivalent to the wedge of n circles corresponding to

each loop xi, thexei generate C1(D

∞

n ) as a free Z[t

±1_{]-module. Then, the boundary}

map

∂ : C1(D∞n ) → C0(D∞n)

sendsx_ei to (t − 1)z. Since De

∞

n is homotopically equivalent to a one-dimensional CW

complex, H1(D∞n ) = ker ∂, which has a basis as Λ-module given by vi =xei+1−xei for i = 1, . . . , n − 1.

The homeomorphism hβ : Dn → Dn can be lifted to an homeomorphism ehβ :

D_n∞→ D∞

n which fixes the fibre over z pointwise. Hence ehβinduces an automorphism

of H1(D∞n ).

Definition 1.2.3. The reduced Burau representation Bt : Bn→ AutΛ(H1(D∞n )) is

the representation of the braid group obtained by sending β to the Λ-linear auto-morphism H1(D∞n ) → H1(D∞n) induced by ehβ.

Fixing the basis v1, . . . , vn as in Lemma 1.2.2, we can view the reduced Burau

representation as a map

Bt: Bn→ GLn−1(Z[t±1]).

Let h , i : H1(Dn∞) × H1(Dn∞) → Z be the alternating algebraic intersection form

obtained by lifting the orientation of Dn to D∞n . Consider

λ : H1(Dn∞) × H1(D∞n ) → Z[t ±1

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CHAPTER 1. L.-T. SIGNATURE AND G.-G. FORMULA 8 defined by λ(x, y) =X j∈Z htj_{x, yit}−j .

Fact 1.2.4. The reduced Burau representation preserves λ.

This is easy to see: in fact since the automorphism induced by ehβ is Λ-linear and

preserves the intersection pairing h , i, then it preserves λ as well. Therefore, given ω ∈ S1_{, the form}

λ(ω) : H1(D∞n ) × H1(D∞n ) → C

obtained by evaluating λ: λ(ω)(x, y) = P

j∈Zhtjx, yiω

−k _{is skew-Hermitian. If we}

think of Bω(α) as a matrix in GLn−1(C), it is unitary with respect to the matrix

representation of λ(ω).

1.2.2 The Gambaudo-Ghys formula

Let H be a complex vector space endowed with a skew-Hermitian intersection form λ. Given two λ-unitary automorphisms γ1, γ2 of H, define the space

Eγ1,γ2 = Im(γ

−1

1 − idH) ∩ Im(idH−γ2).

For x = γ₁−1(x1) − x1 = x2− γ2(x2) ∈ Eγ1,γ2, define b(x, y) = λ(x1+ x2, y). The form

b is well-defined and Hermitian (see [9]).

Definition 1.2.5. In the previous setting, the Meyer cocycle of γ1, γ2is the signature

of the form b and is denoted as Meyer(γ1, γ2).

We are now ready to state the result of Gambaudo and Ghys, which can be found in [9]:

Theorem 1.2.6 (Gambaudo and Ghys). Let α, β be n-braids. Let ω ∈ S1_{\ {1} be}

of order coprime to n. Then

σω( cαβ) − σω(α) − σb ω( bβ) = − Meyer(Bω(α),Bω(β)). This result is very important for at least two reasons:

• it relates two important objects in knot theory: the Levine-Tristram signature and the reduced Burau representation;

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CHAPTER 1. L.-T. SIGNATURE AND G.-G. FORMULA 9 • it provides an efficient algorithm to calculate the Levine-Tristram signature of every link starting from the standard generators of the braid group. This is due to the fact that the Meyer cocycle and the Burau representation can be computed, and that the Levine-Tristram signature of the unlink vanishes. In the following chapters we will see a generalization of the Levine-Tristram sig-nature, called multivariate sigsig-nature, and the aim will be to calculate the defect of the new generalized signature.

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Chapter 2 Multivariate Levine-Tristram

signature

In this chapter we introduce the multivariate signature and we introduce some of its properties. Most of the content of this chapter can be found in [3].

2.1 Coloured links

There are some concepts related to knots that do not extend uniquely to links. For example we might want to endow our link with extra structure and give an ordering to the components: we obtain an ordered link and we say that two such links are isotopic if there is an isotopy that respects the ordering as well. Interpolating between links and ordered links there are coloured links:

Definition 2.1.1. A µ-coloured link L is an oriented link in S3 _{together with a}

surjective map assigning to each component of L a colour in {1, . . . , µ}.

We will often write L = L1t · · · t Lµ where Li is the sublink of the components

of L of colour i.

Remark 2.1.2. A 1-coloured link is an ordinary link, while an ordered link L is a µ-coloured link if µ is equal to the number of connected components of L.

2.1.1 Generalized Seifert surfaces

C-complexes are the generalization of Seifert surfaces for coloured links. They were first introduced by Cooper in [5] and studied by Cimasoni [1].

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CHAPTER 2. MULTIVARIATE LEVINE-TRISTRAM SIGNATURE 11 Definition 2.1.3. A C-complex for a µ-coloured link L = L1t · · · t Lµ is a union

S = S1∪ · · · ∪ Sµ of surfaces in S3 such that:

• for all i the surface Si is a Seifert surface for Li. We allow in this case Si to be

disconnected, but there should be no closed components;

• for all i 6= j, Si and Sj intersect transversely in a possibly empty union of clasp

intersection (see Figure 2.1);

• there are no triple intersections: for i, j, k pairwise distinct, Si ∩ Sj ∩ Sk is

empty.

It is proved in [1] that every coloured links has a C-complex.

Figure 2.1: A clasp intersection.

Figure 2.2: A ribbon intersection. These are not allowed in C-complexes.

2.2 The multivariate signature

In order to define the ordinary Levine-Tristram signature we defined a form, called Seifert form, on the first homology group of a Seifert surface for our link. What we define now is a generalization of the Seifert form using C-complexes.

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CHAPTER 2. MULTIVARIATE LEVINE-TRISTRAM SIGNATURE 12

2.2.1 Generalizing the Seifert form

Let S = S1∪. . .∪Sµbe a C-complex for a µ-coloured link L. Let Ni ∼= Si×[−1, 1] be a

tubular neighbourhood of Si. For εi ∈ {±1} let Siεi denote the surface Si×{εi} ⊂ Ni.

Let Y be the complement of Sµ

i=1int(Ni) in S 3_{. Then, for ε = (ε} 1, . . . , εµ) ∈ {±1}µ_{, set} Sε = µ [ i=1 Sεi i ∩ Y.

Since the intersections are clasps, there is an obvious homotopy equivalence be-tween S and Sε_{, inducing an isomorphism H}

1(S) → H1(Sε). Let

iε : H1(S) → H1(Sε) → H1(S3\ S)

be the composition of this isomorphism with the inclusion induced homomorphism. We are now ready to define

αε: H1(S) × H1(S) → Z

to be the bilinear form

αε(x, y) = lk(iε(x), y).

Fix a basis of H1(S) and let Aε denote the matrix of αε. Notice that when µ = 1,

α−1 is the usual Seifert form. Moreover, for all ε, the identity A−ε = ATε holds.

2.2.2 The multivariate signature

Let ω = (ω1, . . . , ωµ) ∈ T∗µ, where T∗µ∼= (S1\ {1})µ. Set

H(ω) := X ε∈{±1}µ µ Y i=1 (1 − ωεi i )Aε.

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CHAPTER 2. MULTIVARIATE LEVINE-TRISTRAM SIGNATURE 13 Remark 2.2.1. The matrix H(ω) is Hermitian, because H(ω) = H(ω) = H(ω)T_.

Definition 2.2.2. The Levine-Tristram multivariate signature σω(L) of a µ-coloured

link L in ω ∈ Tµ ∗ is

σω(L) := sign(H(ω)).

Theorem 2.2.3. The multivariate signature is a well-defined isotopy invariant of coloured links.

Of course the signature does not depend on the choice of the basis of H1(S), so

it remains to be proved that it does not depend on the C-complex S. In his article [1], Cimasoni provides a set of four moves that transform a C-complex for L into another C-complex for the same link. These moves have the property that when S0 is another C-complex for L, then S0 can be obtained from S by a finite number of these operations. In order to prove the theorem it is sufficient to show that each of these moves preserves the signature. A proof can be found in [3].

2.3 Basic properties of the signature

The following proposition descends directly from the fact that H(ω) = H(ω)T_.

Proposition 2.3.1. For any ω = (ω1, . . . , ωµ) ∈ T∗µ, let ω denote (ω −1

1 , . . . , ωµ−1).

Then

σω(L) = σω(L).

Remark 2.3.2. Let L be a coloured link, and let L0denote the coloured link obtained by reversing the orientation of the sublink Lµ of L. Then if S = S1∪ · · · ∪ Sµ is a

C-complex for L, S0 = S1 ∪ · · · ∪ (−Sµ) is a C-complex for L0. This implies that

HL0(ω₁, . . . , ω_µ−1, ω_µ) = H_L(ω₁, . . . , ω_µ−1, ω−1_µ ). Therefore if −L denotes the coloured

link L with the opposite orientation, then σω(L) = σω(−L) for all ω ∈ T∗µ.

Remark 2.3.3. Let L be a coloured link, and let L denote the mirror image of L. Then if S is a C-complex for L, the mirror image S is a C-complex for L. Therefore Aε

S = −AεS and HL(ω) = −HL(ω) for all ω. Hence σω(L) = −σω(L) for all ω ∈ T∗µ

and the signature can detect chirality.

Corollary 2.3.4. If a coloured link is isotopic to its mirror image σω(L) is identically

zero.

The following proposition descends directly from the fact that, given two links, a C-complex for their disjoint union is given by the disjoint union of the C-complexes of the two links.

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CHAPTER 2. MULTIVARIATE LEVINE-TRISTRAM SIGNATURE 14 Proposition 2.3.5. Let L and J be µ-coloured links. Then, for all ω ∈ Tµ

∗

σω(L t J ) = σω(L) + σω(J ).

2.4 The four-dimensional interpretation

As the Levine-Tristram signature, the multivariate signature admits a 4-dimensional interpretation, studied by Cimasoni and Florens in [3] (Section 6.1) which relies on branched covers.

Start with a C-complex S = S1 ∪ · · · ∪ Sµ for a µ-coloured link L in S3 = ∂D4.

We can assume without loss of generality that the surfaces Si are connected and

we can think of pushing the interior of each Si inside D4, obtaining µ properly

embedded surfaces F1, . . . , Fµ. We can push them so that they only intersect each

other transversely in double points. We call a surface F = F1∪ · · · ∪ Fµ in D4 with

these properties a bounding surface for L.

If we write νF to denote the union of some open tubular neighbourhoods of the Fi, call WF = D4\ νF the exterior of F in D4.

Lemma 2.4.1. Let F = F1∪ . . . ∪ Fµ be the union of µ properly embedded, compact,

connected and oriented surfaces Fi ⊂ D4 which only intersect each other transversely

in double points. Then H1(WF) is freely generated by the meridians of the components

Fi.

Proof. Let Bp denote a small open ball around each intersection point p of F . Let us

denote by F_i◦ the surface Fi without some open discs around the intersection points.

Notice that WF = D4\ G p Bp∪ µ G i=1 ν(F_i◦) ! . The Mayer-Vietoris sequence for X = D4\S

pBp gives · · · H2(X) H1( Fµ i=1F ◦ i × S1) H1( Fµ i=1F ◦ i × D2) ⊕ H1(WF) H1(X) · · ·

Thanks to the K¨unneth formula this simplifies to H1 µ G i=1 S1 ! ∼ = H1(WF),

where each S1 _{corresponds to the meridian of one F} i.

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CHAPTER 2. MULTIVARIATE LEVINE-TRISTRAM SIGNATURE 15 Let L be a µ-coloured link. Fix k1, . . . , kµ positive integers. There is a projection

H1(WF) → Ck1 × · · · × Ckµ

sending the meridian of the i-th surface Fi to a fixed generator ti of the cyclic group

Cki. Consider the cover of WF associated to the kernel of this map. On this cover

there is an action of G := Ck1 × · · · × Ckµ. Using this cover, one can form a G-cover

of D4 branched along F , which we call WF (see [3], section 6.2). If we fix ω1, . . . , ωµ

roots of unity of respective order k1, . . . , kµ, there is a character

χ : G → C∗

sending ti to ωi and extended linearly. Of course, H2(WF; C) is endowed with a

C[G]-module structure. We can therefore consider

H2(WF)χ:= {x ∈ H2(WF; C) | gx = χ(g)x for all g ∈ G}.

The Hermitian intersection form on H2(WF; C) (which is the only Hermitian

ex-tension of the real intersection form to H2(WF; C)) restricted on this space gives a

Hermitian pairing. We call its signature the ω-signature on WF and we denote it by

sign_ω(WF). The following theorem is due to Cimasoni and Florens [3] and gives a

4-dimensional interpretation of the signature.

Theorem 2.4.2. Let k1, . . . , kµ, and G be as above. Let L be a µ-coloured link and

let F be a bounding surface for L. Let WF be the G-fold cover of D4 branched along

F . Then for every ω = (ω1, . . . , ωµ) ∈ T∗µ of orders k1, . . . , kµ the following equality

holds

σω(L) = σω(WF).

This interpretation was used by Cimasoni and Conway in [2] to provide an evalu-ation of the additivity defect of the multivariate Levine-Tristram signature. However this interpretation only works for ω1, . . . , ωµ roots of unity. The main goal of this

thesis is to use a more algebraic interpretation of the signature, studied by Conway, Nagel and Toffoli [6] to calculate the additivity defect for a wider subset of T_∗µ.

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Chapter 3 Twisted homology

Most of the material of this chapter can be found in [7].

3.1 Twisted homology

Let X be a connected CW complex and let Y ⊂ X be a possibly empty subcomplex of X. Let p : eX → X be the universal cover of X and define eY := p−1(Y ).

The fundamental group π1(X) acts on the left on eX and this action gives to

C∗( eX, eY ) := C∗CW( eX, eY ) a left Z[π1(X)]-module structure.

Let R be a ring and let M be a left R-module and a right Z[π1(X)]-module such

that the left and the right action are compatible, that is to say M is a (R, Z1[π1

(X)])-bimodule.

Definition 3.1.1. The chain complex C∗(X, Y ; M ) := M ⊗_Z[π1(X)]C∗( eX, eY ) of left

R-modules is said to be the twisted chain complex with coefficients in M . The correspondent homology left R-modules H∗(X, Y ; M ) are called twisted homology

modules of (X, Y ) with coefficients in M .

Remark 3.1.2. The isomorphism class of H∗(X, Y ; M ) does not depend on the

choice of a basepoint for the fundamental group.

We will denote by Hk(X; M ) the twisted homology modules Hk(X, ∅; M ).

Let us now see that the twisted homology modules of X encompass the cellular homology modules of X and also the one of its regular covering spaces.

More precisely, let ψ : π1(X) → G be a surjective homomorphism and let p :b b

X → X be the regular cover of X associated to ker(ψ). Let us define bY = p_b−1(Y ). In this situation C∗( bX, bY ) has a left Z[G]-module structure.

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CHAPTER 3. TWISTED HOMOLOGY 17 We can extend ψ linearly to a map Ψ : Z[π1(X)] → Z[G]. Hence, every (R,

Z[G])-bimodule M can be extended to a (R, Z[π1(X)])-bimodule just by defining the right

action of Z[π1(X)] as

m · γ := m · Ψ(γ) for every M ∈ M and γ ∈ Z[π1(X)]. We obtain:

Lemma 3.1.3. Let (X, Y ) be a pair where X is a connected CW complex and Y a subcomplex.

Let ψ : π1(X) → G be a surjective group homomorphism and let M be a (R,

Z[G])-bimodule. Hence there exists an isomorphism of chain complexes of left R-modules: M ⊗_Z[π₁(X)]C∗( eX, eY ) ' M ⊗Z[G]C∗( bX, bY ).

As a consequence, we can compute the left R-modules H∗(X, Y ; M ) by means of the

complex M ⊗_Z[G]C∗( bX, bY ).

Proof. The projection π : eX → bX induces

Z[G] ⊗Z[π1(X)]C∗( eX, eY ) → C∗( bX, bY )

g ⊗ α 7→ g · π∗(α)

which we claim is defined and is an isomorphism. Let us see that it is well-defined: for g ∈ Z[G] let γ ∈ Z[π1(X)] be such that g = Ψ(γ). The properties of the

covers imply that g ·π∗(α) = π∗(γ ·α). Moreover, if {αi}i∈I is a basis of Cn( eX, eY ) as a

free Z[π1(X)]-module, then {π∗(αi)}i∈I is a basis of Cn( bX, bY ) as a free Z[G]-module.

Therefore the map is both surjective and injective.

Example 3.1.4. Let ψ : π1(X) → Zµ = ht1, . . . , tµi be an homomorphism and fix

ω = (ω1, . . . , ωµ) ∈ Tµ = (S1\ {1})µ.

The homomorphism ψ can be extended to Ψ : Z[π1(X)] → Z[Zµ] = Z[t±11 , . . . , t±1µ ]

and can be composed with the ring homomorphism Z[t±11 , . . . , t ±1

µ ] → C which

eval-uates t = (t1, . . . , tµ) in ω. We therefore obtain a ring homomorphism

φ : Z[π1(X)] → C

which induces on C a (C, Z[π1(X)])-bimodule structure.

In this situation, to emphasize the choice of ω we write Cω instead of simply C. Therefore the (C, Z[π1(X)])-bimodule structure on Cω allows us to consider the

complex vector spaces Hk(X; Cω). When it is not clear from the context we will

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CHAPTER 3. TWISTED HOMOLOGY 18

3.2 Twisted cohomology

Let R denote a ring endowed with an involution s 7→ s which satisfies: rs = s r,

r + s = r + s, 1 = 1.

Remark 3.2.1. _{• The ring Z[G] has a natural involution, given by the map} G 3 g 7→ g−1 _{extended linearly on Z[G].}

• Complex conjugation on C is an involution.

Given a left (respectively right) R-module N let us indicate with inv(N ) the right (respectively left) R-module which has the same additive group as N but the action of R on N is precomposed with the involution of R.

In this way, given an (R, Z[π1(X)])-bimodule M , both M and inv(C∗( eX, eY )) are

right Z[π1(X)]-modules.

Definition 3.2.2. The cochain complex of left R-modules

C∗(X, Y ; M ) := Hom_Mod−Z[π₁(X)](inv(C∗( eX, eY )), M )

is called the twisted cochain complex of (X, Y ) with coefficients in M . The cor-responding cohomology left R-modules H∗(X, Y ; M ) are called twisted cohomology modules of (X, Y ) with coefficients in M .

3.3 The evaluation map

Let R be a ring with involution and let M, N be two (R, Z[π1(X)])-bimodules.

More-over, let S be an (R, R)-bimodule. Let us consider a sesquilinear pairing

h−, −i : M × N → S

which is also non-singular and π1(X)-invariant, that is to say:

• hm + p, n + qi = hm, ni + hm, qi + hp, ni + hp, qi for all m, p ∈ M and n, q ∈ N ; • hrm, sni = rhm, nis for all m ∈ M , n ∈ N and r, s ∈ S;

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CHAPTER 3. TWISTED HOMOLOGY 19 • for every m ∈ M , m 6= 0, there exists n ∈ N , such that hm, ni 6= 0 and for

every n ∈ N , n 6= 0, there exists m ∈ M such that hm, ni 6= 0; • hmγ, nγi = hm, ni for all m ∈ M , n ∈ N and γ ∈ π1(X).

The proof of the following result can be found in [7] (Lemma 5.4.2).

Lemma 3.3.1. Let (X, Y ) be a CW pair. Let us call for simplicity π := π1(X).

Then the following map is an isomorphism of cochain complexes of left R-modules: κ : Hom_{Mod −Z[π]}(inv(C∗( eX, eY )), M ) → inv(HomR−Mod(N ⊗Z[π]C∗( eX, eY ), S))

f 7→ (n ⊗ σ) 7→ hf (σ), ni. Moreover the following is a homomorphism of left R-modules:

HiinvHomR−Mod(N ⊗Z[π]C∗( eX, eY ), S)

→ inv (HomR−Mod(Hi(X, Y ; N ), S)) .

Remark 3.3.2. Given a ring with involution R, M and N two (R, Z[π1

(X)])-bimodules, an (R, R)-bimodule S and a sesquilinear pairing h−, −i : M × N → S, which is non-singular and π1(X)-invariant, Lemma 3.3.1 provides an evaluation map:

ev : Hi(X, Y ; M ) → inv(HomR−Mod(Hi(X, Y ; N ), S).

3.4 Poincar´

e duality

Let X denote an n-manifold. Pick a triangulation for X and let X0 be the manifold X equipped with the dual cellular decomposition.

Lemma 3.4.1. There is a canonical isomorphism of left π1(X)-modules

Cn−k( eX) ∼

−→ inv(Hom_Z[π₁(X)]−Mod(Ck(fX0, g∂X0), Z[π1(X)]).

This isomorphism assigns to c0 ∈ Cn−k( eX) a map

[−, c0] : Ck(fX0, g∂X0) → Z[π1(X)]

by the formula

[c, c0] := X

γ∈π1(X)

(c · γc0)γ

for all c ∈ Ck(fX0, g∂X0). Here (c · γc0) indicates the integer intersection number of c

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CHAPTER 3. TWISTED HOMOLOGY 20 These isomorphisms are sometimes called universal Poincar´e duality. The proof of this Lemma can be found in [12] (Lemma 1).

The following theorem shows that Poicar´e duality holds for all coefficients: Theorem 3.4.2 (Poincar´e duality). Let X be a compact and oriented n-manifold, let M be an (R, Z[π1(X)])-bimodule. There are isomorphisms:

Hk(X, ∂X; M ) ' Hn−k(X; M )

Hk(X; M ) ' Hn−k(X, ∂X; M ).

Proof. Set π := π1(X), and tensor the universal Poincar´e duality by M :

M ⊗_Z[π]Cn−∗( eX) ∼= M ⊗Z[π]inv(HomZ[π]−Mod(C∗(fX

0_{, g}_∂X0_{), Z[π])}

as chain complexes.

As a consequence, to prove the theorem it is enough to show that there is a left Z[π]-modules isomorphism:

M ⊗_Z[π]inv(Hom_Z[π]−Mod(C∗(fX0, g∂X0), Z[π]) ∼= HomMod −Z[π](inv(C∗(fX0, g∂X0)), M ).

Let us show this in two steps. First note that there is a canonical isomorphism of left Z[π]-modules between

inv(Hom_Z[π]−Mod(Ck(fX0, g∂X0), Z[π]))

and

Hom_{Mod −Z[π]}(inv(C∗(fX0, g∂X0)), Z[π])

thanks to Lemma 3.3.1 with R = M = N = S = Z[π] and the sesquilinear pairing M × N → S given by (p, q) 7→ pq.

Moreover since inv(C∗(fX0, g∂X0)) is finitely generated and free as a right Z[π]-module,

there is a canonical isomorphism of left Z[π]-modules between M ⊗_Z[π]Hom_{Mod −Z[π]}(inv(C∗(fX0, g∂X0)), Z[π])

and

Hom_{Mod −Z[π]}(inv(C∗(fX0, g∂X0)), M ⊗_Z[π]Z[π]). The combination of these homomorphisms concludes our proof.

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CHAPTER 3. TWISTED HOMOLOGY 21

3.5 Twisted intersection pairings

Let W be a manifold of even dimension 2k. Let R be a ring with involution, let M be a (R, Z[π1(W )])-bimodule and let S be a (R, R)-bimodule.

Moreover, let

h−, −i : M × M → S

be a non-singular and π1(W )-invariant sesquilinear pairing.

The inclusion j : (W, ∅) ,→ (W, ∂W ) induces j∗ : Hk(W ; M ) → Hk(W, ∂W ; M ).

Recall that the Poincar´e duality provides an isomorphism PD : Hk(W, ∂W ; M ) → Hk(W ; M )

and that Remark 3.3.2 gives us an evaluation map

ev : Hk(W ; M ) → inv(HomR−Mod(Hk(W ; M ), S)).

Let us denote by Φ the composition Φ : Hk(W ; M ) j∗ −→ Hk(W, ∂W ; M ) PD −−→ Hk_{(W ; M )}₋ev_{→ inv(Hom} R−Mod(Hk(W ; M ), S))

Definition 3.5.1. The intersection pairing

λM(W ) : Hk(W ; M ) × Hk(W ; M ) → S

defined by

λM(W )(x, y) = Φ(y)(x)

is called twisted intersection pairing.

Remark 3.5.2. The pairing λM(W ) is Hermitian when k is even, it is skew-Hermitian

otherwise. This derives directly by the properties of the isomorphism in Lemma 3.4.1 There is no reason for λM(W ) to be non-singular. In fact it is easy to see that

the image of the map H1(∂W ; M ) → H1(W ; M ) annihilates λM(W ).

Example 3.5.3. Let W be a 2k-manifold with boundary, µ a positive integer. As in Example 3.1.4, let ψ : π1(W ) → Zµ = ht1, . . . , tµi be an homomorphism and

let ω ∈ Tµ _{:= (S}1 _{\ {1})}µ_{. Consider the linear extension of ψ and evaluate it in}

(t1, . . . , tµ) = ω. We obtain a homomorphism φ : Z[π1(W )] → C of rings which

preserves the natural involutions seen in Remark 3.2.1. Notice that Cω× Cω → C

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CHAPTER 3. TWISTED HOMOLOGY 22 is a non-singular, π1(W )-invariant, sesquilinear pairing.

We can therefore consider

λ_Cω(W ) : H_k(W ; Cω) × H_k(W ; Cω) → C.

If k is even, then the intersection pairing is also Hermitian and we can consider its signature.

Definition 3.5.4. The signature of the intersection form λ_Cω(W ) is called ω-twisted

signature of W and we indicate it by

sign_ω(W ) := sign(λ_Cω(W )).

3.6 Long exact sequence of the pair

Let X be a connected CW complex and let Y ⊂ X be a possibly empty connected subcomplex of X. Let p : eX → X indicate the universal cover of X and define Y := p−1(Y ).

Let y ∈ Y . The group π1(X, y) acts on the left on eX and this action endows

C∗(Y ) with a structure of left Z[π1(X)]-module. Given a ring R and an (R, Z[π1

(X)])-bimodule M , define

H∗(Y ⊂ X; M ) := H∗(M ⊗_Z[π1(X)]C∗(Y )).

Since

0 → C∗(Y ) → C∗( eX) → C∗( eX, Y ) → 0

is exact and the Z[π1(X)]-modules involved are all free, it follows that

0 → M ⊗_Z[π₁(X)]C∗(Y ) → M ⊗_Z[π1(X)]C∗( eX) → M ⊗Z[π1(X)]C∗( eX, Y ) → 0

is an short exact sequence of chain complexes, and we obtain a long exact sequence of left R-modules:

· · · → Hk(Y ⊂ X; M ) → Hk(X; M ) → Hk(X, Y ; M ) → Hk−1(Y ⊂ X; M ) → · · ·

We would like however to obtain a more classic long exact sequence of the pair, using H∗(Y ; M ) instead of H∗(Y ⊂ X; M ). To do this, we first have to endow M with

a (R, π1(Y ))-bimodule structure. Notice that the inclusion induced homomorphism

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CHAPTER 3. TWISTED HOMOLOGY 23 Theorem 3.6.1. There is an isomorphism

M ⊗_Z[π₁(Y,y)]C∗( eY ) → M ⊗_Z[π1(X,y)]C∗(Y ),

where eY indicates the universal cover of Y . This map just depends on the choice of basepoints as lifts of y in eY and Y .

It immediately follows that H∗(Y ⊂ X; M ) is isomorphic to H∗(Y ; M ).

Proof. Suppose y ∈ Y ⊂ X is the fixed basepoint. Let _ey ∈ eY be a lift of y and y a lift of y in Y . Notice that eY → Y lifts to a unique map eY → Y that sends e

y 7→ y. The free Z[π1(Y )]-module Ck( eY ) is generated by one lift of each k-cell of Y .

We fix and call these lifts _ee1, . . . ,eem. Similarly, the free Z[π1(X)]-module Ck(Y ) is generated by one lift of each k-cell of Y . The image of the _eei’s along eY → Y gives a

basis ef1, . . . , efm of Ck(Y ).

The map

M ⊗_Z[π1(Y )]Ck( eY ) → M ⊗Z[π1(X)]Ck(Y )

m ⊗_eei 7→ m ⊗ efi

is well-defined and clearly both surjective and injective.

Corollary 3.6.2. In this context there is a long exact sequence of the pair: · · · → Hk(Y ; M ) → Hk(X; M ) → Hk(X, Y ; M ) → Hk−1(Y ; M ) → · · ·

Remark 3.6.3. Notice that we can adapt the proof above to have an interpretation of the long exact sequence of the pair when Y is disconnected. In the case Y =F

iYi

is disconnected we can interpret Hk(Y ; M ) asL_iHk(Yi; M ) and with this definition

we still have a long exact sequence of the pair:

· · · → Hk(Y ; M ) → Hk(X; M ) → Hk(X, Y ; M ) → Hk−1(Y ; M ) → · · ·

3.7 Other properties of twisted homology

Let f : Y → X be a continuous map between two connected CW complexes. Let M have the structure of a (R, π1(X))-bimodule. The map

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CHAPTER 3. TWISTED HOMOLOGY 24 determines also a (R, π1(Y ))-bimodule structure on M . In this case f induces a

homomorphism of left R-modules

f∗ : H∗(Y ; M ) → H∗(X; M ).

It is therefore easy to see that two homotopic maps Y → X induce the same (R, π1(Y ))-bimodule structure on M and as a consequence the map f∗ : H∗(Y ; M ) →

H∗(X; M ) depends only on the homotopy class of f .

3.7.1 Excision and the Mayer-Vietoris sequence

Twisted homology verifies other classical properties of cellular homology, such as excision: let A and B be two subcomplexes of X such that A ∪ B = X. Let pX : eX → X be the universal cover of X and pB : eB → B the universal cover of

B. Then Ck( eB, p−1_B (B ∩ A)) is generated as a Z[π1(B)]-module by one lift of each

k-cell of B which do not belong to A. We call these lifts _ee1, . . . ,eem. Let b ∈ B ⊂ X be the basepoint, let eb be a lift of b in eB and let bb be a lift of b in eX. There is a unique lift eB → p−1_X (B) which sends eb → bb. Similarly, Ck( eX, p−1_X (A)) is generated as

a Z[π1(X)]-module by one lift of each k-cell of X which do not belong to A, which

are still m. We call these lifts ef1, . . . , efm, and we can suppose that each efi is exactly

the image of _eei via the map eB → p−1_X (B). Then the inclusion induced map:

M ⊗_Z[π₁(B)]Ck e B, p−1_B (B ∩ A)→ M ⊗_Z[π₁(X)]Ck e X, p−1_X (A) m ⊗e_ei 7→ m ⊗ efi

is clearly both surjective and injective. Hence we obtain excision:

Hk(B, B ∩ A; M ) ∼= Hk(X, A; M ).

Notice that the Mayer-Vietoris sequence is actually a consequence of excision and the long exact sequence of the pair: let in fact A and B be two subcomplexes of X such that A ∪ B = X. We have the following commutative diagram:

· · · Hn+1(B, B ∩ A; M ) Hn(B ∩ A; M ) Hn(B; M ) Hn(B, B ∩ A; M ) · · ·

· · · Hn+1(X, A; M ) Hn(A; M ) Hn(X; M ) Hn(X, A; M ) · · ·

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CHAPTER 3. TWISTED HOMOLOGY 25 and it is a purely algebraic fact that such a diagram with every third vertical map an isomorphism gives rise to a long exact sequence involving the remaining non-isomorphic terms as follows:

· · · → Hn+1(X; M ) → Hn(B ∩ A; M ) → Hn(A; M ) ⊕ Hn(B; M ) → Hn(X; M ) → · · ·

Example 3.7.1. It is easy to verify that the twisted homology of the point is

Hk({∗}; M ) =

(

M if k = 0; 0 otherwise.

This fact combined with the homotopy invariance of twisted homology allows us to compute the twisted homology of every contractible space.

Example 3.7.2. Let us now consider the twisted homology of the sphere Sn _for

n > 0. The n-sphere has a CW decomposition with one 0-cell and one n-cell. For n > 1, this easily implies that

Hk(Sn; M ) =

(

M if k = 0, n; 0 otherwise. .

For n = 1 the situation is not so straightforward, in fact the chain complex M ⊗_Z[π₁(S1_)]C_∗(S1) is non-zero only in the 0-th and 1-st position:

0 → M ⊗_Z[π₁(S1_)]C₁(fS1) id ⊗∂ −−−→ M ⊗_Z[π₁(S1_)]C₀(fS1) → 0 where M ⊗_Z[π₁(S1_)]C₁(fS1) ∼= M ⊗ Z[π1(S1)]C0(fS 1_{) ∼}_{= M. Hence} Hk(S1; M ) =      ker(id ⊗∂) if k = 1; M/Im(id ⊗∂) if k = 0; 0 otherwise.

There is no reason for the map id ⊗∂ to be zero and in fact we will later see a case where this map is an isomorphism and therefore Hk(S1; M ) = 0 for every k. As a

consequence we realize that twisted homology does not always behave as homology with coefficients: for instance there are examples of path connected spaces whose 0-th twisted homology group is not M .

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CHAPTER 3. TWISTED HOMOLOGY 26 Example 3.7.3. Let T be a torus. There is a cell decomposition of T with one 0-cell p, two 1-cells which we identify with the meridian m and the longitude l, and one 2-cell e. Let p be a lift of p in the universal cover e_e T and lift m to m, l to e_e l and e to e

e. The twisted chain complex with coefficients in M is 0 → M id ⊗∂2

−−−→ M × M id ⊗∂1

−−−→ M → 0.

Of course (id ⊗∂2)(n⊗ee) = n⊗(m+m·ee l−l·m−l), while (id ⊗∂e 1)(n⊗m) = n⊗(m·e p−e p)e and (id ⊗∂1)(n ⊗ el) = n ⊗ (l ·ep −p).e

The following proposition will be needed later on:

Proposition 3.7.4. Let X be a compact, oriented and connected 2k+1-manifold. We denote by Σ = ∂X the boundary of X. Let R be a ring endowed with an involution, let M be an (R, Z[π1(X)])-bimodule and S an (R, R)-bimodule. Note that the inclusion

induced map π1(Y ) → π1(X) determines an (R, π1(Y ))-bimodule structure over M .

Then we have that the R-module

L := kerHk(Σ; M ) i

−

→ Hk(X; M )

is totally isotropic with respect to the twisted intersection form over Hk(Σ; M ), that

is to say

L ⊂ Ann L = {x ∈ Hk(Σ; M ) | λM(Σ)(v, x) = 0 for all v ∈ L}.

If the evaluation map is an isomorphism for all CW complex pairs, then L is La-grangian, that is to say L = Ann(L).

Proof. The following diagram commutes:

Hk+1(X, Σ; M ) Hk(X; M ) inv(HomR−Mod(Hk(X; M ), S)) Hk(Σ; M ) Hk(Σ; M ) inv(HomR−Mod(Hk(Σ; M ), S)) Hk(X; M ) Hk+1(X, Σ; M ) inv(HomR−Mod(Hk+1(X, Σ; M ), S)) ∂ PD ev −◦i PD i ev −◦∂ PD ev

Consider α, β ∈ L. Since they belong to ker(i), the long exact sequence of the pair ensures that there exists B ∈ Hk+1(X, Σ; M ) such that ∂B = β in Hk(Σ; M ).

Hence

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CHAPTER 3. TWISTED HOMOLOGY 27 because i(α) = 0.

Suppose now the evaluation maps are all isomorphisms.

Fix an element γ ∈ Hk(Σ; M ) \ L. Then i(γ) 6= 0 ∈ Hk(X; M ). The image

of i(γ) in inv(HomR−Mod(Hk+1(X, Σ; M ), S)) is non-zero, hence there exists C ∈

Hk+1(X, Σ; M ) such that

0 6= ev ◦ PD(i(γ))(C) = ev ◦ PD(γ))(∂C) = λM(Σ)(∂C, γ).

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Chapter 4 Novikov-Wall non-additivity

theorem

When two manifolds M1 and M2 have diffeomorphic boundary we can glue them

along their boundary to obtain another manifold W . If the dimension is a multiple of 4, on the manifolds there is a well-defined symmetric real intersection form and the Novikov additivity theorem says that the signature of W is equal to the sum of the signatures of M1 and M2; hence we say that the signature is additive. On

the other hand, if we glue the two manifolds along a submanifold of the boundary instead, the signature of the resulting manifold will not be the sum of the signatures of M1 and M2 anymore. However the Novikov-Wall non-additivity theorem provides

an evaluation of the defect of additivity of the signature.

In this section we will formulate the statement of Novikov-Wall non-additivity theorem in the twisted case and we will give a proof of this statement, adjusting when needed the proof from Wall in [16] for the untwisted case.

4.1 The Maslov index

The Maslov index, in the sense that was used by Wall in its article [16], associates an integer to three isotropic subspaces of a symplectic real vector space. In this section we review this construction and adapt it to the setting of skew-Hermitian complex vector spaces.

Let (H, λ) be a finite dimensional skew-Hermitian complex vector space and let L1, L2 and L3 be three isotropic subspaces of H. Consider the Hermitian form f

defined on (L1+ L2) ∩ L3 as follows: for a, b ∈ (L1+ L2) ∩ L3, write a = a1+ a2 with

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CHAPTER 4. NOVIKOV-WALL NON-ADDITIVITY THEOREM 29 ai ∈ Li for i = 1, 2 and set

f (a, b) = λ(a2, b).

Remark 4.1.1. The form f is a well-defined Hermitian form. In fact if a = a1+a2 =

a0₁+ a0₂ and b = b1+ b2, then

λ(a2, b) − λ(a02, b) = λ(a2− a02, b) = λ(a2− a02, b1) = λ(a − a, b1) = 0.

Moreover it is Hermitian:

f (b, a) = λ(b2, a) = λ(b, a) − λ(b1, a)

but λ(b, a) = 0 since a, b ∈ L3. Therefore

f (b, a) = λ(a, b1)

and λ(a, b1) is easily seen to be equal to f (a, b).

Definition 4.1.2. The signature of f is called the Maslov index of L1, L2 and L3.

It will be denoted by Maslov(L1, L2, L3).

The following Lemma states some important properties of the Maslov index and is straightforward to prove.

Lemma 4.1.3. The Maslov index satisfies the following properties. 1. If L1, L2, L3 are isotropic subspaces of H and L01, L

0 2, L

0

3 are isotropic subspaces

of H0, then Li⊕ L0i for i = 1, 2, 3 is an isotropic subspace of H ⊕ H 0 _and

Maslov(L1⊕ L01, L2⊕ L02, L3⊕ L03) = Maslov(L1, L2, L3) + Maslov(L01, L 0 2, L

0 3).

2. For any isotropic subspaces L1, L2 ⊂ H,

Maslov(L1, L2, L2) = 0.

3. If L1, L2, L3 are isotropic subspaces of H and ψ is a unitary automorphism of

H, then ψ(L1), ψ(L2), ψ(L3) are isotropic subspaces of H and

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CHAPTER 4. NOVIKOV-WALL NON-ADDITIVITY THEOREM 30 Notice that f (a, b) vanishes whenever a or b belong to (L1∩ L3) + (L2∩ L3). Let

σ be a permutation of (1, 2, 3). Another important property of the Maslov index is Maslov(L1, L2, L3) = sgn(σ) Maslov(Lσ(1), Lσ(2), Lσ(3)).

In fact it is clear that if we exchange L1 and L2 the Maslov index switches sign.

Moreover there is an isomorphism (L1+ L2) ∩ L3

(L1∩ L3) + (L2∩ L3)

→ (L2+ L3) ∩ L1 (L2∩ L1) + (L3∩ L1)

defined by sending a = a1 + a2 to a1. It preserves the respective Hermitian forms

and hence the Maslov index.

4.2 Novikov-Wall non-additivity

Let Y be an oriented connected compact 4k-manifold and let X0 be an oriented

compact (4k − 1)-manifold, properly embedded into Y so that ∂X0 = X0 ∩ ∂Y .

Suppose that X0 splits Y into two manifolds Y− and Y+. For ε = ±, denote by Xε

the closure of ∂Yε \ X0, which is a compact (4k − 1)-manifold. Let Z denote the

compact (4k − 2)-manifold

Z = ∂X0 = ∂X+ = ∂X−.

The manifolds Y+ and Y− inherit an orientation from Y . Orient X0, X+ and X−

such that

∂Y+ = X+∪ (−X0)

and

∂Y− = X0∪ (−X−)

and orient Z such that

Z = ∂X−= ∂X+= ∂X0.

Suppose that we are given a homomorphism

ψ : π1(Y ) → Zµ= ht1, . . . , tµi.

By Example 3.1.4 this endows C with a (C, Z[π1(Y )])-bimodule structure, given

by the map Ψ : Z[π1(Y )] → C, and we emphasize the choice of ω by indicating this

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CHAPTER 4. NOVIKOV-WALL NON-ADDITIVITY THEOREM 31 π1(S) → π1(Y ) induces on C a (C, Z[π1(S)])-bimodule structure, which we still

indicate as Cω. Hence there are twisted homology modules with coefficients in Cω. Recall that for manifolds W with even dimension we also have a twisted intersection pairing λ_Cω(W ) and that in the case the dimension is a multiple of 4 the intersection

pairing is Hermitian, and therefore it has a well-defined signature sign_ω(W ).

Figure 4.1: The manifold Y in the setting of the Novikov-Wall non-additivity theo-rem.

Theorem 4.2.1 (Novikov-Wall non-additivity theorem). In the situation above, sign_ω(Y ) = sign_ω(Y+) + signω(Y−) + Maslov(L−, L0, L+)

where Lε = ker(H2k−1(Z; Cω) → H2k−1(Xε; Cω)) for ε = −, +, 0.

Notice that L+, L−, L0 are subspaces of H2k−1(Z; Cω) which is a skew-Hermitian

complex vector space with intersection pairing λ_Cω(Z). Moreover, by Proposition

3.7.4, we see that the spaces Lε are indeed totally isotropic.

Proof. Consider the exact sequence H2k(∂Y ; Cω) → H2k(Y ; Cω)

i

−

→ H2k(Y, ∂Y ; Cω) → H2k−1(∂Y ; Cω).

By Proposition 7.5.4 in [7] (it is a consequence of the universal coefficient spectral sequence for twisted coefficients), the evaluation map

ev : H2k_{(W ; C}ω)−→ inv(Hom∼ _C(H2k(W ; Cω), C))

is an isomorphism for all CW complexes W . Therefore the radical of λ_Cω(Y ) coincides

with the kernel of the map i, which is the image of H2k(∂Y ; Cω) → H2k(Y ; Cω). For

this reason the signature of λ_Cω(Y ) coincides with the signature of the associated

non-singular form on G2k(Y ) := Im(H2k(Y ; Cω) → H2k(Y, ∂Y ; Cω)). Notice that

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CHAPTER 4. NOVIKOV-WALL NON-ADDITIVITY THEOREM 32 We choose analogous splittings for Yε with ε = ±

H2k(Yε) ∼= Im(H2k(∂Yε; Cω) → H2k(Yε; Cω)) ⊕ G2k(Yε).

Lemma 4.2.2. The subspace of H2k(Y, ∂Y ; Cω) orthogonal to

Im(H2k(Y−; Cω) ⊕ H2k(Y+; Cω) ,→ H2k(Y ))

is the image of H2k(X0, Z; Cω).

Proof. Let jε : H2k(Yε; Cω) → H2k(Y ; Cω) be the inclusion induced map. The map

kε : H2k(Y, ∂Y ; Cω) ∼= inv(HomC(H2k(Y ; Cω), C)) −◦jε

−−→ inv(Hom_C(H2k(Yε; Cω), C))

is such that for y ∈ H2k(Y ; Cω) and yε∈ H2k(Yε; Cω) we get

λ_Cω(Y )(y, j_ε(y_ε)) = k_ε(i(y))(y_ε).

The subspace orthogonal to Im(jε) with respect to the pairing

H2k(Y ; Cω) × H2k(Y, ∂Y ; Cω) → C

(x, y) 7→ ev(PD(y))(x) is the kernel of kε, hence of

H2k(Y, ∂Y ; Cω) → H2k(Yε, ∂Yε; Cω) ∼= inv(HomC(H2k(Yε; C ω

), C)).

Hence the subspace orthogonal to Im(H2k(Y−; Cω) ⊕ H2k(Y+; Cω) ,→ H2k(Y )) is

ker(k−) ∩ ker(k+), that is the kernel of the map

H2k(Y, ∂Y ; Cω) → H2k(Y−, ∂Y−; Cω) ⊕ H2k(Y+, ∂Y+; Cω) ∼= H2k(Y, X; Cω)

where X = X+∪ X− ∪ X0 and the last isomorphism is due to the Mayer-Vietoris

sequence of the pair. The obtained map

H2k(Y, ∂Y ; Cω) → H2k(Y, X; Cω)

is actually equal to the one induced by inclusion. Thanks to the relative Mayer-Vietoris sequence for the pairs (Y, ∂Y ) and (X, X)

ker(k−) ∩ ker(k+) = Im(H2k(X, ∂Y ; Cω) → H2k(Y, ∂Y ; Cω))

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CHAPTER 4. NOVIKOV-WALL NON-ADDITIVITY THEOREM 33 The inclusion induced maps H2k(Yε; Cω) → H2k(Y ; Cω) preserve the twisted

pair-ings, therefore the image of G2k(Yε) via this map is contained in G2k(Y ). Since the

pairing is non-singular on G2k(Yε) the two maps for ε = ± are injective, and their

images are clearly orthogonal to each other. For this reason their images are in direct sum.

Let K denote the orthogonal complement of the image of G2k(Y−) ⊕ G2k(Y+) in

G2k(Y ). Notice that

sign_ω(Y ) − sign_ω(Y+) − signω(Y−)

is the signature of the intersection pairing restricted to K. The image of H2k(∂Y−) ⊕ H2k(∂Y+) H2k(Y−) ⊕ H2k(Y+) H2k(Y ) H2k(Y, ∂Y )

is clearly contained in G2k(Y ) and more precisely in K. Here we avoided specifying

the twisted coefficients Cω. Let us call S the image of

H2k(∂Y−; Cω) ⊕ H2k(∂Y+; Cω) → H2k(Y, ∂Y ; Cω),

which is a subset of G2k(Y ). If S⊥ denotes the orthogonal complement of S in K,

then S⊥ coincides with the orthogonal complement in G2k(Y ) of the image of the

map

H2k(Y−; Cω) ⊕ H2k(Y+; Cω) → H2k(Y, ∂Y ; Cω).

This is implied by the splitting

H2k(Yε; Cω) ∼= Im (H2k(∂Yε; Cω) → H2k(Yε; Cω)) ⊕ G2k(Yε).

By the previous lemma we obtain

S⊥ = G2k(Y ) ∩ Im(H2k(X0, Z; Cω) → H2k(Y, ∂Y ; Cω)).

Clearly the triangle of inclusions

∂(Yε, ∅) (Yε, Xε)

(Y, ∂Y )

commutes, and, thanks to excision, H∗(Yε, Xε; Cω) ∼= H∗(X0, Z; Cω), therefore S ⊂

S⊥. Now let L be a a complement of S in S⊥ S⊥= S ⊕ L.

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CHAPTER 4. NOVIKOV-WALL NON-ADDITIVITY THEOREM 34 The orthogonal complement of S⊥in K is S, hence the radical of the form restriction to S⊥ must be S. This implies that the form is non-singular on L.

Since it is non-singular on K, it is also non-singular on the orthogonal complement L⊥ of L in K, and K = L ⊕ L⊥.

Notice that S ⊂ L⊥ and, since the form is non-singular on K, that the orthogonal complement of S in L⊥ coincides with S⊥∩ L⊥ _{= (S ⊕ L)}⊥_{= (S}⊥₎⊥ _{= S.}

This means that dim S = 1₂dim L⊥ and it is a standard argument that the dimension of a maximal totally isotropic subspace of a non-singular Hermitian vector space is the minimum between the index of positivity and negativity.

Therefore the signature of our intersection form restricted to L⊥ is 0. Hence the signature of the form on K is the same as the signature of the form on L and on S⊥ since S⊥ = S ⊕ L and S is the radical of the form on S⊥. Recall that L was defined as a complement of

S = Im(H2k(∂Y−; Cω) ⊕ H2k(∂Y+; Cω) → H2k(Y, ∂Y ; Cω))

in

S⊥= G2k(Y ) ∩ Im(H2k(X0, Z; Cω) → H2k(Y, ∂Y ; Cω))

where

G2k(Y ) = Im(H2k(Y ; Cω) → H2k(Y, ∂Y ; Cω)).

We can lift any element x ∈ S⊥ to y ∈ H2k(X0, Z; Cω) (non uniquely). We can then

pick z = ∂y ∈ H2k−1(Z; Cω). The map χ : S⊥→ H2k−1(Z; Cω) such that χ(x) = z is

a well-defined homomorphism: suppose y ∈ H2k(X0, Z; Cω) such that the image of y

in H2k(Y, ∂Y ; Cω)) is 0. It is enough to show that y maps to z = 0 in H2k−1(Z; Cω).

By excision

H2k(X0, Z; Cω) ∼= H2k(X, ∂Y ; Cω).

Therefore y comes from an element w ∈ H2k+1(Y, X; Cω) thanks to the relative

Mayer-Vietoris sequence for the pairs (Y, ∂Y ) and (X, X). Note that z is also the image of w via

H2k+1(Y, X; Cω) → H2k(X; Cω) → H2k(X, Z; Cω) → H2k−1(Z; Cω)

which is the zero map, hence z = 0.

We claim χ is a map of S⊥ onto L0∩ (L−+ L+) and the kernel is a subspace of

S, where L−, L+ and L0 are the ones defined in the statement of the Theorem.

The preimage of S⊥ in H2k(X0, Z; Cω) is the set of y ∈ H2k(X0, Z; Cω) whose

im-age in H2k(Y, ∂Y ; Cω) is also in the image of H2k(Y ) or equivalently maps to 0 in

H2k−1(∂Y ; Cω). Thus the image of S⊥ in H2k−1(Z; Cω) is

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CHAPTER 4. NOVIKOV-WALL NON-ADDITIVITY THEOREM 35 The first space is L0 by the exact sequence of the pair, the second one is

Im(H2k(∂Y, Z; Cω) = H2k(X−, Z; Cω) ⊕ H2k(X+, Z; Cω) → H2k−1(Z; Cω)

where the first equality follows from the relative Mayer-Vietoris sequence of the pairs (X+, Z) and (X−, Z). It easily follows that the second member is actually L−+ L+.

This means that S⊥ projects onto L0 ∩ (L−+ L+). Suppose now x ∈ S⊥ such that

z = ∂y = 0. There is w ∈ H2k(X0; Cω) which maps onto y. The diagram

H2k(X0; Cω) H2k(∂Yε; Cω)

H2k(X0, Z; Cω) H2k(Y, ∂Y ; Cω)

commutes, therefore x is in the range of H2k(∂Y−; Cω) ⊕ H2k(∂Y+; Cω), hence it

belongs to S.

Since the kernel of χ is contained in the radical of our form, we can push-forward the form.

Let a ∈ L0∩ (L−+ L+) and suppose α ∈ H2k(X0, Z; Cω) is such that ∂α = a. Let

a− ∈ L− and a+ ∈ L+ be such that a = a−+ a+.

Notice that there are β ∈ H2k(X+, Z; Cω) γ ∈ H2k(X−, Z; Cω) such that ∂β = a+and

∂γ = a−. Let η be a cycle representing α, i.e. η ∈ Cω⊗π1(X0)C2k( fX0, eZ) ∼= C

ω_⊗ π1(Y )

C2k(π−1(X0), π−1(Z)) and let ξ, ζ be cycles representing −β, −γ respectively. Here

π is the universal cover π : eY → Y .

An element of S⊥ which maps to a is represented by η. The cycle η + ξ + ζ is a closed cycle in Y and maps onto η via the inclusion Y ,→ (Y, ∂Y ). Let a0 ∈ L0∩ (L−+ L+)

be another element and let ξ0, η0, ζ0 be the corresponding system of cycles. Let us call c = η + ξ + ζ =P

i(ηi+ ξi+ ζi) ⊗eei whereeei are lifts of the 2k-cells in Y and in the dual cell decomposition c0 = η0+ ξ0+ ζ0 = P

j(η 0

j+ ξj0 + ζj0) ⊗ee

0

j. We shall pick

the lifts _eei in such a way that two of them intersect the same connected component

of π−1(Z) if the correspondent cells in Y intersect the same connected component of Z. Then their intersection is

X g∈π1(Y ) X i,j (ηi+ ξi+ ζi)(ηj0 + ξ 0 j+ ζ 0 j)(eei· gee 0 j)Ψ(g)

where Ψ : Z[π1(Y )] → C was the map which induced on Cω its bimodule structure

and (− · −) is the integer intersection number. We can push the _ee0_j’s that are in X−

(and therefore are in the expression of gζ0) and in X0 (and therefore are in the

ex-pression of gη0) slightly inside π−1(Y−) ⊂ eY and the rest of it (the ones corresponding

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CHAPTER 4. NOVIKOV-WALL NON-ADDITIVITY THEOREM 36

Figure 4.2: The blue cycle represents c and the red one represents the slight defor-mation of gc0.

Then λ_Cω(Y )(c, c0) just depends on the intersections of the terms coming from η

and gξ0 for g ∈ π1(Y ), and is equal to

X g∈π1(Y ) X i,j (_eei· gee 0 j)ηiξ0jΨY(g). Note that (_eei· gee 0 j) is equal to −(eei· g∂ee 0

j)X0, the intersection number in π

−1_(X 0)

of these two cells when g∂_ee0_j is in π−1(X0) ∩ π−1(X+) = π−1(Z). Here the minus sign

comes from the boundary orientation. Therefore, this intersection number is equal to −(∂_eei· g∂ee

0

j)Z in π−1(Z). This number can be different from zero only if ∂eei and g∂_ee0_j lie in the same connected component of π−1(Z), and this can happen only if g ∈ π1(Y ) is in the range of π1(Z).

Therefore, since ΨZ factors as π1(Z) → π1(Y ) ΨY −−→ C, we get λ_Cω(Y )(c, c0) = −λ Cω(Z)(∂η, ∂ξ 0 ) = −λ_Cω(Z)(∂a, −∂a0₊) = −λ Cω(Z)(∂a+, ∂a 0 ). Therefore sign(S⊥) = − Maslov(L−, L+, L0) = Maslov(L−, L0, L+).

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Chapter 5 The isotropic functor

5.1 The category of coloured tangles

Braids can be seen as a particular example of tangles. Intuitively, a tangle is a par-ticular properly embedded 1-dimensional submanifold of D2_{×[0, 1] with a prescribed}

oriented boundary. More precisely, given a positive integer n, let p(n)_j be the point 2j − n − 1

n , 0

for j = 1, . . . , n. Let ε and ε0 be sequences of ±1’s of length n and n0 respectively. Definition 5.1.1. A (ε, ε0)-tangle is an oriented, smooth, properly embedded 1-submanifold of D2× [0, 1] whose oriented boundary is

n0 X j=1 ε0_j(pn_j0, 1) − n X j=1 εj(pnj, 0).

Figure 5.1: A (ε, ε0)-tangle, where ε = (−1, +1, +1) and ε0 = +1 37

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CHAPTER 5. THE ISOTROPIC FUNCTOR 38 An oriented tangle τ is called µ-coloured if each of its components is assigned a label in {1, . . . , µ}. We will call a µ-coloured (ε, ε0)-tangle a (c, c0)-tangle, where c and c0 are the sequences of ±1, ±2, . . . , ±µ induced by the orientation and the colouring of the tangle.

Two (c, c0)-tangles τ1 and τ2 are isotopic if there exists an self-homeomorphism

h of D2_{× [0, 1] that keeps the boundary ∂(D}2_{× [0, 1]) fixed and such that h sends}

τ1 homeomorphically to τ2 preserving the orientation and the colouring. We denote

by Tµ(c, c0) the set of isotopy classes of (c, c0)-tangles. Let idc indicate the isotopy

class of the trivial (c, c)-tangle {p(n)₁ , . . . , p(n)n } × [0, 1]. Given a (c, c0)-tangle τ1 and a

(c0, c00)-tangle τ2 such that they both meet the boundary of D2× [0, 1] orthogonally,

their composition is the (c, c00)-tangle τ2τ1obtained by gluing the two D2×[0, 1] along

the disk corresponding to c0 and then shrinking the height of the resulting D2_{× [0, 2]}

by a factor 2.

Figure 5.2: The composition of two coloured tangles. Here τ1 is a (c, c0)-tangle with

c = (−2) and c0 = (−1, +1, −2), while τ2 is a (c0, c00)-tangle with c00 = (−1, +1, −2).

Clearly, the composition of tangles induces a composition Tµ(c, c0) × Tµ(c0, c00) → Tµ(c, c00)

on the isotopy classes of µ-coloured tangles.

Definition 5.1.2. All finite sequences c of elements in {±1, ±2, . . . , ±µ} as ob-jects, and the isotopy classes of (c, c0)-tangles as morphisms c → c0 form a category

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CHAPTER 5. THE ISOTROPIC FUNCTOR 39 Tangles_µ called the category of µ-coloured tangles.

Definition 5.1.3. Given an endomorphism in Tangles_µ, that is to say τ ∈ Tµ(c, c),

one can define its closure as the µ-coloured link _bτ ⊂ S3 _{obtained from τ adding}

n oriented coloured parallel strands in S3 _{\ (D}2 _{× [0, 1]) connecting p}(n)

j × {1} to

p(n)_j × {0} for j = 1, . . . , n. Here n denotes the length of the sequence c.

Figure 5.3: The closure of a tangle.

5.2 The isotropic and lagrangian categories

We introduce the category IsotrΛof isotropic relations over a ring Λ. Fix an integral

domain Λ endowed with a ring involution a 7→ a.

Definition 5.2.1. A skew-Hermitian Λ-module H is a finitely generated Λ-module endowed with a possibly degenerate skew-Hermitian form λ.

Given a skew-Hermitian Λ-module H, −H will indicate the same module endowed with the opposite form −λ.

Definition 5.2.2. Given a submodule V of a skew-Hermitian Λ-module H, the annihilator of V is the submodule

An extension of a theorem by Cimasoni and Conway

An extension of a theorem by

Cimasoni and Conway

Candidata:

Relatore:

Alice Merz

Prof. Paolo Lisca

Contents

Introduction

Chapter 1

Levine-Tristram signature and the

formula of Gambaudo and Ghys

1.1

The four-dimensional interpretation

1.2

The Gambaudo-Ghys formula

1.2.1

The reduced Burau representation

1.2.2

The Gambaudo-Ghys formula

Chapter 2

Multivariate Levine-Tristram

signature

2.1

Coloured links

2.1.1

Generalized Seifert surfaces

2.2

The multivariate signature

2.2.1

Generalizing the Seifert form

2.2.2

The multivariate signature

2.3

Basic properties of the signature

2.4

The four-dimensional interpretation

Chapter 3

Twisted homology

3.1

Twisted homology

3.2

Twisted cohomology

3.3

The evaluation map

3.4

Poincar´

e duality

3.5

Twisted intersection pairings

3.6

Long exact sequence of the pair

3.7

Other properties of twisted homology

3.7.1

Excision and the Mayer-Vietoris sequence

Chapter 4

Novikov-Wall non-additivity

theorem

4.1

The Maslov index

4.2

Novikov-Wall non-additivity

Chapter 5

The isotropic functor

5.1

The category of coloured tangles

5.2

The isotropic and lagrangian categories