Global Surgery Formula for the Casson-Walker Invariant. (AM-140), Volume 140

by Christine Lescop

Overview

This book presents a new result in 3-dimensional topology. It is well known that any closed oriented 3-manifold can be obtained by surgery on a framed link in S
3. In Global Surgery Formula for the Casson-Walker Invariant, a function F of framed links in S
3 is described, and it is proven that F consistently defines an invariant, lamda (l), of closed oriented 3-manifolds. l is then expressed in terms of previously known invariants of 3-manifolds. For integral homology spheres, l is the invariant introduced by Casson in 1985, which allowed him to solve old and famous questions in 3-dimensional topology. l becomes simpler as the first Betti number increases.


As an explicit function of Alexander polynomials and surgery coefficients of framed links, the function F extends in a natural way to framed links in rational homology spheres. It is proven that F describes the variation of l under any surgery starting from a rational homology sphere. Thus F yields a global surgery formula for the Casson invariant.

Product Details

ISBN-13:	9781400865154
Publisher:	Princeton University Press
Publication date:	09/08/2014
Series:	Annals of Mathematics Studies , #140
Sold by:	Barnes & Noble
Format:	eBook
Pages:	150
File size:	8 MB

About the Author

Christine Lescop is Researcher in Mathematics at the Centre National de la Recherche Scientifique at the Institut Fourier in Grenoble, France.

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Global Surgery Formula for the Casson-Walker Invariant

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By Christine Lescop

PRINCETON UNIVERSITY PRESS

Copyright © 1996 Princeton University Press
All rights reserved.
ISBN: 978-0-691-02133-1

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CHAPTER 1

Introduction and statements of the results

§1.1 Introduction

In 1985, A. Casson defined an integer invariant for oriented integral homology 3-spheres by introducing an appropriate way of counting the conjugacy classes of the SU(2)-representations of their fundamental group. He proved that his invariant λc satisfies the following interesting properties:

λc vanishes on homotopy spheres,

λc is additive under connected sum,

λc changes sign under orientation reversal,

a simple surgery formula describes the variation of λc under a surgery on a knot transforming an integral homology sphere into another one, and,

λc lifts the Roblin μ-invariant from Z/2Z to Z (recall that if M is a Z-homology 3-sphere, 8μ(M) is the signature (mod 16) of any smooth spin 4-manifold with boundary M).

This last property allowed Casson to answer several old and well-known questions about the Roblin invariant and the topology of 3-manifolds (see [G-M2] or [A-M]).

In 1988, K. Walker extended the Casson invariant, and all of its original properties, to rational homology 3-spheres; furthermore, he gave a combinatorial and elementary defmition for his extension (see [W]).

According to a theorem independently proved by Lickorish and Wallace (see [Rou]), any compact connected oriented 3-manifold without boundary can be presented by a surgery diagram (or a framed link) in S3. A theorem of Kirby describes simple moves which suffice to relate two surgery presentations of the same 3-manifold (see [Kir 2]).

This book states and proves a global surgery formula for the Casson-Walker invariant, that is it describes explicitly a function F of the surgery diagrams presenting rational homology spheres such that F gives the Casson-Walker invariant of the manifolds presented by such diagrams.

This function F extends naturally to all surgery diagrams in S3. In Chapter 3, it is verified directly to be invariant under the Kirby moves; it therefore defines an invariant λ of closed oriented 3-manifolds.

The function F also extends to surgery diagrams in any rational homology sphere and provides, as shown in Chapter 4, a surgery formula describing the variation of λ under any surgery starting from a rational homology sphere. This surgery formula generalizes the Walker one-component surgery formula.

Chapter 5 describes the invariant λ as a function of previously known invariants for manifolds with nonzero first Betti number. λ becomes simpler as the first Betti number increases, vanishing for manifolds with first Betti number greater than 3.

These results are the main results of this book. They are precisely stated in §1.5, after the description of F in § 1.4, which involves notation introduced in §1.2 and §1.3. Section 1.6 outlines the proofs of these results and refers to the following chapters for details.

The function F is defined for any rational surgery presentation in any rational homology sphere. In the case of an integral surgery presentation, the signature of the associated 4-dimensional cobordism is part of the function F. This yields a straightforward comparison (Proposition 6.3.8) between λ and the Roblin μ-invariant; this also allows us to give a new proof that μ is well-defined in §6.3.C.

§6.1 applies the surgery formula to the computation of λ for all oriented Seifert fibered spaces, as an example.

§1.4 describes F as a sum of a combination D of certain derivatives of several variable Alexander polynomials, and a function of linking numbers associated with the presentations.

The part D of F bad been found by S. Boyer and D. Lines in [B-L 1], where they showed in particular that the function of surgery presentations of integral homology spheres (λcoχ - D) depends only on the homotopical type of the link and on its framing.

(Here χ denotes the function mapping a surgery presentation to its associated 3-manifold.)

§1.7 gives two more definitions for F. Definition 1.7.8 describes F, for surgery presentations with null-homologous components, as a function of one-variable Alexander polynomials and linking numbers of the presentations.

The corresponding surgery formula generalizes the Hoste surgery formula for the variation of the Casson invariant under surgeries with diagonal linking matrices (see [Hos]).

The main (and only) tool in this book is the normalized several-variable Alexander polynomial. All its required properties are stated in Chapter 2 and proved in the appendix.

Acknowledgements

I am grateful to Steven Boyer, Daniel Lines and Kevin Walker whose articles [B-L 1] and [W] inspired this one.

I thank Christian Blanchet, Michel Boleau, Lucien Gullou, Nathan Habegger, Pierre Vogel and especially Alexis Marin. They had enough courage to start reading the first version of this book and their remarks were of much use in the rewriting process.

My thanks also go to Viviane Baladi, Nathalie Hunter-Mandon and Lisa Ramig for their generous help with my English.

Part of this work has been written when I was a visitor at "l'Université du Québec à Montréal". I warmly acknowledge the hospitality of this University and last but not least the kindness of Steven BoYER during my stay there.

I also thank the referees for their thorough reading of this book and for their appropriate suggestions.

§1.2 Conventions

• The boundary of an oriented manifold is oriented with the "outward normal first" convention; unless otherwise specified, when a manifold and its boundary are oriented, the orientations are supposed to agree.

• An integral (respectively rational) homology sphere is a closed 3-manifold with the same Z-homology (respectively (Q-homology) as the usual 3-sphere S3.

• Curves or surfaces are identified with their homology classes when it does not seem to cause confusion.

• In any closed oriented 3-manifold M, the linking number LkM(J,K) of two disjoint oriented links J and K representing 0 in H1(M;(Q) is defined as follows:

If J represents 0 in H1(M) = H1(M;Z) and if ΣJ is an oriented Seifert surface with boundary J, then LkM(J,K) is the algebraic intersection number of ΣJ and K in M. LkM{.,K) is next extended by (Q -linearity on Ker(H1(M\K;Q) [right arrow] H1 (M;(Q)).

The linking number LkM(.,.) is symmetric.

• The oriented meridian of an oriented knot K in an oriented 3-manifold is the meridian m(K) of K which links K positively.

• If Γ is an abelian group, |Γ| denotes its order. (If Γ is finite, |Γ| is its cardinality, otherwise |Γ| is zero.)

• If K is a knot in a rational homology sphere M, OM(K) denotes the order of the class of Kin H1(M):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

• Let x be an element of IR, sign(x) = x/|x| = ±1 if x is nonzero and sign(0) = 0.

§1.3 Surgery presentations and associated functions

Definition 1.3.1:

A primitive satellite of K is an oriented simple nonseparating closed curve of the boundary [partial derivative]T(K) of a tubular neighborhood T(K) of K.

If K is an oriented knot embedded in a rational homology sphere M and if ll is a primitive satellite of K, the homology class [μ] of μ in [partial derivative]T(K) will be identified with the ordered pair (p,q) of (QxZ, where

• the curve μ is homologous to qK in T(K),

• p = LkM(μ,K), (p must then be congruent to qLkM(K,K) modulo Z, where LkM(K,K) [member of] Q/Z denotes the self-linking number of K in M, that is the linking number of Kanda parallel of K mod Z).

If μ is not a meridian of K, ±[μ] will be identified with the rational number p/q.

Definition 1.3.2: (Surgery presentation)

A surgery presentation in an oriented rational homology sphere M is a link L, each component Ki of which is oriented and equipped with a primitive satellite Jli, specified by a pair (pi, qi) with a positive qi.

The manifold presented by such a surgery presentation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is denoted by χ(L) and defined in the following way:

Let T(Ki) be a tubular neighborhood of Ki in M, and let hi be a homeomorphism from the boundary (S1 × S1)i of (D2 × S1)i to the boundary [partial derivative]T(<Ki) of T(Ki) which sends the meridian (S1 × {*})i to the curve μi of [partial derivative]T(Ki), then χ(L) is the following closed 3-manifold:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The surgered manifold χ(L) inherits the orientation of M.

L is said to be an integral surgery presentation if all the qi are equal to 1.

The curve μi is called the characteristic curve of the surgery on Ki and ({0} × S1)i is called the core of the surgery performed on Ki. (The characteristic curve μi is a meridian of the core of the surgery performed on Ki in the surgered manifold.)

Definitions 1.3.3: (Some junctions ofthe surgery presentations)

Let L be a surgery presentation in a rational homology sphere M as in 1.3 .2. Let lij be defmed by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

• The linking matrix of L is defined by

E(L) = [lij]1 ≤ i, j ≤ n

• Note that the matrix F(L) = [qj lik]1 ≤, j ≤ n is a presentation matrix of H1(χ(L)) if M is an integral homology sphere, and that, in any case:

1.3.4 |H1(M)| |det(F(L))| = |H1(χ(L))|

F(L) is called the framing matrix of the presentation L.

• The sign of the surgery presentation L, denoted by sign(L ), is equal to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where b-(L) denotes the number of negative eigenvalues of the symmetric matrix E(L).

Note the following equality coming from 1.3.4,

1.3.5 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

• We will denote by signature(E(L )) the signature of E(L) (that is, (b+ L) - b-(L)), where b+(L) denotes the number of positive eigenvalues of E(L)).

• Restriction of a surgery presentation

If I is a subset of N={1, ..., n}, then LI (respectively LI) denotes the surgery presentation obtained from L (respectively the link obtained from L) by forgetting the components whose subscripts do not belong to I.

Unless otherwise specified, L will denote the surgery presentation of Definition 1.3.2 and we will use the notation introduced in §1.3 for L.

§1.4 Introduction of the surgery formula F

The Alexander series D is, up to a change of variables (see 2.1.1), the normalized Alexander polynomial in several variables Δ, and it will be precisely defined in Chapter 2 as an invariant of oriented links in oriented rational homology spheres.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The surgery formula will only require the following coefficient of the Alexander series:

Definition 1.4.1: The ζ-coefficient of a link in an oriented rational homology sphere

Let L = (Ki)i [member of] {1, ..., n} be an oriented link in an oriented rational homology sphere, then ζ(L) denotes the following coefficient:

• If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

• If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

According to the properties of the Alexander series (see §2.3), the ζ-coefficient of a link L of n components in an oriented rational homology sphere M satisfies:

1.4.2 ζ(L) depends neither on the orientation of the components of L nor on their order.

1.4.3 ζ(L [subset] (-M)) = (-l)n-1 ζ(L [subset] M). ((-M) denotes the manifold M after an orientation reversal.)

1.4.4 If L is a split link, ζ(L) is zero.

The following definition can be skipped by the reader interested only in integral surgery presentations.

Definition 1.4.5: Dedekind sums (see [R-G])

Let p be an integer; let q be an integer or the (mod p)-congruence class of an integer, also denoted by q; the Dedekind sum s(q,p) is the following rational number:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

First definition of F (or first description of the surgery formula) (see 1.7.3 for an equivalent similar definition of F and 1.7.8 for a definition of F from one-variable polynomials)

1.4.6 The 8-linking of LI, denoted by L8 (LI), is defined by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where σI denotes the set of bijections from {1, ..., i =#I} to I.

Later on, the 8-linking of LI will be seen as a sum running over all combinatorial ways of identifying the elements of I with the set of vertices of a graph G whose underlying space is the figure eight drawn in Figure 1.1. The summand corresponding to a graph G (whose vertices are labelled by I) is the linking of L with respect to this graph (see Definition 2.4.2).

1.4.7The modified B-linking of LI, denoted by M8 (LI), is defined by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

1.4.8The function FM is defined on the set of all surgery presentations in a rational homology sphere M, with values in (Q, by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Recall that the determinant of an empty matrix equals one.

If L is an integral surgery presentation, the qi's are one, thus the Dedekind sums s(., qi) are zero.

When M is an integral homology sphere, the LkM(Ki, Ki) are integers and can be ignored.

Recall also that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

§1.5 Statement of the theorem

Theorem

T1 There exists a function λ from the set of oriented closed 3-manifolds (up to orientation-preserving homeomorphisms) to 1/12 Z, defined by:

For any surgery presentation Lin S3,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The invariant λ satisfies the following properties:

T2 For any rational homology sphere M and for any surgery presentation lH in M,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

T3 If (-M) denotes the manifold M equipped with the opposite orientation, and if β1 (M) denotes the dimension of H1(M;Q):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

T4 If # denotes the connected sum,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

T5 Let M be a closed oriented 3-manifold.

T5.0 If β1 (M) = 0, (i.e., if M is a rational homology sphere), and if λw denotes the Walker invariant as described in [W], then

λ(M) = |H1(M)|/2 λw(M)

(If M is an integral homology sphere and if λc denotes the Casson invariant as described in [G-M2], then

λ(M) = λc(M))

(Continues...)

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Excerpted from Global Surgery Formula for the Casson-Walker Invariant by Christine Lescop. Copyright © 1996 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents

Ch. 1	Introduction and statements of the results	5
Ch. 2	The Alexander series of a link in a rational homology sphere and some of its properties	21
Ch. 3	Invariance of the surgery formula under a twist homeomorphism	35
Ch. 4	The formula for surgeries starting from rational homology spheres	60
Ch. 5	The invariant [lambda] for 3-manifolds with nonzero rank	81
Ch. 6	Applications and variants of the surgery formula	95
Appendix: More about the Alexander series	117
Bibliography	147
Index	149