On the Casson Invariant



Christine Lescop

March

Abstract

The Casson invariant is a top ological invariant of closed oriented

manifolds It is an integer that counts the SU representations of the

fundamental group of these manifolds in a sense intro duced by Casson

in Its rst prop erties allowed Casson to solve famous problems in

dimensional top ology

The Casson invariant can also b e indep endently dened in a com

binatorial way as a function of Alexander p olynomials of framed links

presenting the manifolds It has numerous interesting prop erties It b e

haves nicely under most top ological mutations such as orientation reversal

connected sum surgery regluing along surfaces This makes it easy to

compute and to use The Casson invariant contains the Rohlin invariant of

Z

homology spheres that is the signature of any smo oth spin manifold

2Z

b ounded by such a sphere It is also explicitly related to quantum invari

ants and is the rst nite typ e invariant in the sense of Ohtsuki This

Ohtsuki notion of nite typ e invariants for manifolds is analogous to the

Vassiliev notion for knots

This talk will b e a general presentation of the Casson invariant where

its newest prop erties and developments will b e emphasized

Keywords Casson invariant manifolds surgery Reidemeister torsion Alexander

p olynomial Torelli group

AMS sub ject classication N M M



CNRS Institut Fourier UMR

Intro duction

The Casson invariant is the simplest top ological invariant of manifolds among

the invariants intro duced after It interacts with many domains of mathe

matics Casson dened it using the dierential top ology of SU representation

spaces It can b e extended in the spirit of this denition using symplectic ge

ometry W CLM C It can also b e dened using innitedimensional analysis

and gauge theory Ta it is the Euler characteristic of the Flo er homology BD

Its natural connections with the structure of the mapping class group have

b een investigated in a series of articles of S Morita Mo who interpreted it

as a certain secondary invariant asso ciated with the rst characteristic class of

surface bundles Its relationship with quantum invariants Mu links it to quan

tum physics and the Dedekind sums which often show up in the study of its

top ological prop erties show that it even interacts with arithmetic

Here with the aim of b eing as elementary as p ossible we will present

a combinatorial denition of the Casson invariant and describ e most of its

top ological prop erties

Surgery presentations of manifolds

Here all the manifolds are compact and oriented unless otherwise mentioned

Boundaries are oriented with the outward normal rst convention We usually

work with top ological manifolds but since any top ological manifold of dimension

less or equal than has exactly one C structure see Ku we will sometimes

make incursions in the smo oth category Manifolds are always considered up to

oriented homeomorphism and emb eddings and homeomorphisms are considered

up to ambient isotopy

Given a manifold M a knot K of M that is an emb edding of the circle

1

S in the interior of M and a parallel of K that is a closed curve on the

b oundary of a tubular neighb orho o d T K of K which runs parallel to K we

can dene the manifold M K obtained from M by surgery on K with

respect to by the following construction remove the interior of T K from

2 1

M and replace it by another solid torus D S glued along the b oundary

2 1

T K of T K by a homeomorphism from T K to D S which maps

2

to D fg

2 1

M K M n T K D S

T (K ) D S

Note that M K is welldened Indeed it is obtained by rst gluing

M n T K along an annulus around in T K and by next a thickened disk to

2 3

lling in the resulting sphere S in the b oundary by a standard ball B

K is called a framed knot A collection of disjoint framed knots is called

a framed Surgery on framed links is the natural generalization of surgery on

framed knots The surgery is p erformed on each comp onent of the link A rst

motivation for taking this op eration in consideration is the following theorem

which is proved in a very elegant way in Ro

Theorem Lickorish Li Wallace W Any closed ie compact

connected without boundary manifold can be obtained from the standard

3

sphere S by surgery on a framed link

We now dene linking numb ers to help parametrizing surgeries Let J

and K b e two disjoint knots in a manifold M which are rationnally null

homologous ie null in H M Q Then there is a closed surface emb edded

1

in M n T K whose b oundary lies in T K and is homologous to dK in T K

for some nonzero d Z and the linking number of J and K l k J K is

unambiguously dened as the algebraic intersection numb er of J and divided

out by d It is symmetric

For R Z Q or ZZ a manifold with the same R homology H R

3

as S is called an R sphere When M is a Qsphere the isotopy class in T K

of the characteristic curve of the surgery is sp ecied by the linking numb er

of and K in M and the framed knot K is also denoted by K l k K

3 3

In particular a framed link in S is a link of S each comp onent of which is

equipp ed by an integer

Figure A surgery presentation of the Poincare sphere see R

A second motivation for studying surgeries is the Kirby calculus which

relates two surgery presentations of the same manifold Following is the Fenn

and Rourke version of the Kirby calculus

Theorem FennRourke FR Kirby K Any two framed links

3

of S presenting the same manifold can be obtained from each other by a nite

number of FRmoves wrt the fol lowing description of FRmoves

3

Let L b e a framed link in S such that a comp onent U of L is a trivial

knot U equipp ed with a parallel satisfying l k U Consider a

U U

2 1

3

cylinder I D emb edded in S n T U so that I S is emb edded in T U

3

Let b e the homeomorphism of S n T U which is the identity outside the

cylinder and which twists the cylinder around its axis so that is mapp ed

U

to the meridian of U Clearly L n U presents the same manifold as L do es

where we think of framed links as links equipp ed with curves to give a meaning

to L n U We dene a FRmove as the op eration describ ed ab ove which

transforms L into L n U or its inverse It is easy to see that such a move do es

not change the presented manifold

According to the ab ove theorem in order to dene an invariant of closed

manifolds it suces to nd a function of surgery presentations invariant under

FRmoves For lack of go o d candidates this pro cess had not b een used b efore

Since with the invasion of quantum invariants there is a lot of manifolds

invariants which have b een proved to b e invariant using this simple principle

RT W but for most of them a top ological interpretation is still to b e

found Here I prop ose rst to intro duce such an invariant function and next

to give the top ological interpretation of the invariant of manifolds it yields a

generalization of the Casson invariant

A combinatorial denition of the Casson in

variant

In order to intro duce our invariant function F we need some notation Let L

K b e a framed link in a Qsphere M K K K l k K

i iN i i i i i i

N f ng is the set of indices of the comp onents of L For a subset I of N

L K E L l k K denotes the symmetric linking

I i iI ij i j ij =1 n

+

matrix of L b L resp b L is the numb er of negative resp p ositive

+

eigenvalues of E L signatureE L b L b L For a Zmo dule A jAj

denotes the order of A that is its cardinality if A is nite and otherwise Note

that

b (L)

jH M Lj detE LjH M j

1 1

Unless otherwise mentioned the homology co ecients are the integers

Now we can set

X

b (L)

detE L L F L

I M

N nI

I N I =

signatureE L

jH M Lj

1

with

I

L L L jH M j L

8 I I 1 I

where L L and L are describ ed b elow

8

L L is the following homogeneous p olynomial in the co ecients of the

8

linking matrix Let G b e a graph whose vertices are indexed by N for an edge

e of G whose ends are indexed by i and j we set l k L e Next we dene

ij

l k L G as the pro duct running over all edges e of G of the l k L e Now

L L is the sum of the l k L G running over all graphs G whose vertices are

8

the elements of N and whose underlying spaces have the form of a gure eight

made of two oriented distinguished circles North and South with one common

vertex

The co ecient can b e dened from the severalvariable Alexander p oly

nomial as dened and normalized in Ha and BL or in Section b elow

for several comp onent links and from the classical Alexander p olynomial of

knots which is the order of the H of the innite cyclic covering of M n K viewed

1

1

as a natural Zt t mo dule normalized in such a way that and is

symmetric

n



n1

L if n

t t

n

L

(1) O (K )

1 M

K if n

1

2jH (M )j 24 O (K )

M

where O K jH M jjTorsionH M n K j is the order of the class

M 1 1 1 1

of K in jH M j It is one if M is a Zsphere

1 1

We can now state the theorem

Theorem L There exists a rational topological invariant of closed

3

manifolds such that for any framed link L in S

3

S L F L

S

The sodened invariant satises the more general surgery formula

Prop erty For any framed link H in a Qsphere M

jH M Hj

1

M H M F H

M

jH M j

1

The principle of the pro of of the theorem is very simple According to the

Fenn and Rourke version of the Kirby theorem it suces to show the invariance

of F under a FRmove and the function F is a function of homological invariants

of the exterior of the framed link whose variation under a homeomorphism of

this exterior can b e followed with some combinatorial eorts See L

The pro of of the general surgery formula rests on the same remark Take

a surgery presentation L of the Qsphere M By transversality we may assume

that H is disjoint of the link L made of the cores of the new solid tori glued

3

by the surgery Then the surgery presentation H can b e seen in S Again we

think of it as a link equipp ed with characteristic curves and the equality to b e

shown is

jH M Hj

1

3

F L F H F H S L

M

S S

jH M j

1

where b oth sides are functions of homological invariants of

3

S n H L M n H L

equipp ed with the surgery curves

Because of the form of the surgery formula it is easy to compare the

invariant with the Rohlin invariant for ZZspheres Before stating the com

parison prop erty let us give a denition of this invariant discovered in A

spin structure on a smo oth manifold of dimension greater or equal than is a

homotopy class of trivializations of its tangent bundle over its skeleton See

Mi for other denitions The Rohlin invariant of a ZZsphere M is the

signature mo d of the intersection form on the H of a smo oth spin ie

2

equipp ed with a spin structure manifold b ounded by M

Prop erty For any ZZsphere M

M jH M jM mo d

1

3

To a surgery presentation L S of a manifold M we may asso ciate

the following natural manifold W b ounded by M W is constructed from

L L

4 2 2

the standard dimensional ball B by gluing handles D D to each com

3 4

p onent of the tubular neighb orho o d of L T L S B with resp ect to

the trivialization given by the characteristic curves which allows to identify

2 1 2 2

a comp onent of T L to D S D D W is next smo othed in a

L

standard way The linking matrix E L is the matrix of the intersection form

on H W wrt the basis of H W asso ciated to its handle decomp osition

2 L 2 L

ab ove A necessary and sucient condition for W to b e spin is that the di

L

agonal of E L is even see GM p and this can always b e realized by

FR moves see Ka In this case it is easy to check that when detEL is

o dd that is when M is a ZZsphere jdetE LjF L signatureE L

S

b elongs to Z This proves the congruence with the Rohlin invariant stated

ab ove and a few classical easy arguments show that this also gives a pro of of

the original Rohlin theorem asserting that the signature of a closed smo oth spin

manifold is divisible by this Rohlin theorem yields the welldenedness of

the Rohlin invariant as a direct corollary see L Sec

The following prop erties of the invariant can also b e checked very easily

Prop erty For any closed manifold M the invariant of the manifold M

obtained from M by orientation reversal satises

(M )+1

M M

where M is the rst Betti number of M

1

Prop erty For any two closed manifolds M and M the invariant of

1 2

def

3 3

M n B M n B satises their connected sum M M

1 2 1 2

S

M M jH M jM jH M jM

1 2 1 2 1 1 1 2

But the main prop erty of is that it can b e expressed in terms of previously

known invariants

Prop erty Let M be a closed manifold

If M then

1

M

1

If M let a b c be a basis of H M and let denote the cup

1

product then

2

M jTorsionH M ja b cM

1

If M let F G be a basis of H M represent it by two closed

1 2

surfaces F and G embedded in general position in M cal l their oriented

intersection cal l the paral lel of wrt the trivialization of the normal

bund le of induced by F and G

M jTorsionH M jl k

1

If M let M be the of M that is again

1

the order of the H of the innite cyclic covering of M viewed as a nat

1

1

ural Zt t module normalized in such a way that M and

1

M t M t

M jTorsionH M j

1

M

If M ie if M is a Qsphere then M is the CassonWalker

1

invariant of M More precisely if M is a Zsphere M is the Casson

invariant of M as normalized in AM GM and in general if denotes

W

the normalization of the Walker invariant used in W

jH M j

1

M M

W

It is now time to describ e the Casson invariant of Zspheres as intro duced

by Casson in

The Casson invariant after Casson

Let M b e a Zsphere A Casson dened M as an algebraic numb er of conju

gacy classes of irreducible SU representations of M in the following way

1

Details can b e found in AM or GM

As any closed manifold M can b e decomp osed into two handleb o dies A

and B glued along a genus g surface A B A handleb o dy is a regular

neighb orho o d of a wedge of circles in a manifold Such a decomp osition M

A B is called a Heegaard splitting of M



For a top ological space X call R X the space of SU representations

of the discrete group X equipp ed with the compact op en top ology The

1

subspace of R X consisting of irreducible representations is an op en set in R X

denoted by RX When X is a free group of rank g for example when

1

X A or B R X has a natural smo oth structure which makes it dieomorphic

g

3 g

S R also has a natural smo oth structure Namely call to SU

the surface obtained from by removing an op en disk cho ose a basep oint of

3

on and call R S the evaluation of a representation of R at

1

The restriction of to R is a submersion Thus R R

b ecomes a natural smo oth g submanifold of R Let X A B or

the free smo oth action of SO SU f g by right conjugation on

R X identies R X with the total space of a principal SO bundle whose

base is a smo oth op en manifold denoted by R X R X is the space of conjugacy

classes of irreducible SU representations of X

1

The inclusions of into A and B identify R A and R B with submani

folds of R and the Van Kamp en theorem identies R M with R A R B

Since M is a Zsphere the only reducible SU representation of M is

1

the trivial one and it can b e shown that R A and R B intersect transver

0

sally at Thus RA R B and hence R A R B are compact Therefore

0

an isotopy with compact supp ort p erturbing the inclusion of R A into R

can make RA transverse to R B inside R Now since RA and R B

are of complementary dimension in R their intersection is a nite numb er

of p oints which can b e given signs or once R A R B and R are

oriented The sum of these signs is denoted by RA R B It is up

^

R()

to sign twice the Casson invariant

In order to suppress the sign indetermination we must sp ecify orientations

SU R A R B and R are oriented arbitrarily SO is oriented by the

double covering SU SO R is oriented as the b er of with the

convention base f iber Once R X is oriented R X is oriented as the base

of a SO bundle with the convention base f iber It can b e shown that the

classical algebraic intersection numb er R A R B is

R( )



Now with all the notations ab ove we can state Cassons original denition

of

g

R A R B

^

R()

M

R A R B

R( )



Casson proved the invariance of using the ReidemeisterSinger theorem

which asserts that two Heegaard splittings of the same manifold b ecome iso

morphic after a nite numb er of stabilizations that are connected sums with

3

the genus one Heegaard splitting of S and following the transformation of the

ab ove denition under a stabilization

Cassons theorem was

Theorem Casson There exists an integral topological invariant

of Zspheres such that

If the trivial representation is the only representation of M into SU

1

then M

M M

M M M M

1 2 1 2

For any knot K in a homology sphere M for any

K M K M

M M mo d

The immediate corollaries of this theorem The Rohlin invariant of a ho

motopy sphere is nul l and The Rohlin invariant of an amphicheral Zsphere

is nul l answered two longunsolved questions in lowdimensional top ology and

allowed Casson to show the existence of a top ological manifold which cannot

b e triangulated see AM as a simplicial complex

Note that the rst assertion of the theorem is a direct corollary of Cassons

denition of The second and third assertions can also b e proved very easily

3

from this denition Since any Zsphere can b e obtained from S by a sequence

of surgeries on knots framed by see GM and b ecause an analogous

the fth assertion surgery formula was known for the Rohlin invariant

8

is a direct consequence of the surgery formula Thus the only diculty in the

pro of of Cassons theorem in addition to inventing this denition is to prove

the surgery formula from the denition ab ove

To prove the surgery formula Casson proved the following lemma A bound

ary link is a link whose comp onents b ound disjoint surfaces in the ambient

3

manifold T denotes the trefoil knot in S pictured in Figure

Lemma Let be a rational invariant of Zspheres such that For any

component boundary link L in a Zsphere M whose components are framed by

X

M L

I

I f12g

Then

3 3

M K M K S T S

3

Then he computed S T and proved that satised the hy

p othesis of the lemma from his denition In fact it is p ossible GM to compute

the Casson invariant of Seifert b ered Zspheres with exceptional b ers and

the variation of the Casson invariant under a surgery along a knot b ounding

an unknotted genus one Seifert surface directly from Cassons denition Both

3

computations give S T

Remark In Lin X S Lin proved that the signature of a knot can also

b e obtained by counting some SU representations of the of its exterior

1

a la Casson Like Cassons comparison of his representation numb er with the

Rohlin invariant Lins pro of that his representation numb er coincides with the

signature is not direct In b oth cases it would b e interesting to have a more

direct identication

Note also that Lemma provides a nice characterization of the Casson

invariant In the same spirit it can b e shown L

Prop erty Any two Zspheres which have the same Casson invariant can

be obtained from one another by a sequence of surgeries on knots with trivial

Alexander polynomial framed by

In K Walker W used the stratied symplectic structure of the rep

resentation spaces Go to give a complete generalization of Cassons work to

Qspheres In this case reducible representations can not b e ignored and basic

dierential top ology do es not suce anymore to provide a p owerful generaliza

tion A weaker generalization of the Casson invariant to Qspheres had b een

prop osed by S Boyer and A Nicas BN Furthermore K Walker gave a very

nice pro of based on Kirby calculus that his onecomp onent surgery formula gives

a consistent denition of his invariant

W

Next S Capp ell R Lee and E Miller CLM generalized Walkers de

nition to other Lie groups like SU n but they have not yet found interesting

prop erties for their invariants C Curtis C studied the SO U S pin

and SO invariants more precisely and proved that they are functions of the

Walker SU invariant

Of course the combinatorial extension of the Casson invariant describ ed

in Section is also a development of Cassons work Indeed without Walkers

generalization of the Casson theorem ab ove and BoyerLiness work BL the

author would not have b een able to nd the general surgery formula of Prop

erty In their work indep endent from Walkers S Boyer and D Lines gave

a combinatorial denition of the restriction of the Walker invariant to ho

W

mology lens spaces they proved a twocomp onent surgery formula formula for

the Casson invariant they exhibited the rst part F of the surgery function F

1

3

the combination of the co ecients and they proved that S F

1

is invariant under link homotopy It must also b e mentioned that the surgery

formula for algebraically split links that are links whose comp onents do not

algebraically link each other is due to Hoste Ho to close this section ab out the

Casson work and some of its developments

Further top ological prop erties of the Casson

invariant

Since the AlexanderConway p olynomial is a wellundersto o d knot invariant

it is easy to apply the surgery formula satised by in order to compute the

invariant of any manifold presented by surgery L L in order to study

the b ehaviour of under other top ological mutations as in D Ki Wo or in

Prop erties and describ ed b elow or in order to compare with other

invariants as H Murakami did to prove that the Walker invariant is equal to a

function of the Reshetikhin and Turaev invariants that he appropriately dened

Mu

Remark In O T Ohtsuki generalized Murakamis work and renormal

ized the Reshetikhin and Turaev invariants into an invariant series of Qspheres

whose rst co ecients are jH j and It would b e interesting to know whether

1

the other co ecients of this series are related to Cassontyp e invariants To study

his series Ohtsuki O dened the notion of nite typ e invariant for Zspheres

This notion is analogous to the notion of Vassiliev invariants of knots Say that

a rational invariant of Zspheres is of AStyp e resp of Btyp e less or equal

than n if for any n comp onent algebraically split resp b oundary link L

in a Zsphere M whose comp onents are framed by

X

M L

I

I f1n+1g

Note that Cassons lemma proves that the Btyp e invariants are

exactly the degree p olynomials in while the Hoste surgery formula Ho

shows that is of AStyp e In fact it is proved that the AStyp e is always a

multiple of and it is conjectured proved that the two mentioned notions of

nite typ e invariants coincide and that the AStyp e is three times the Btyp e

It is not hard to see that for any integer n a degree n p olynomial in is an

invariant of Btyp e n and of AS typ e n Thus the p olynomials in are nice

prototyp es for nite typ e invariants But T T Q Le proved Le that they

are not the only ones It would b e interesting to place the SU ninvariants of

Capp ell Lee and Miller among these nite typ e invariants

It is worth mentioning the existence of some variants of the surgery formula

Prop erty that have not yet b een mentioned to avoid intro ducing to o many

notations Note that in the surgery denition we do not need the characteristic

curve of the surgery to b e parallel to the knot K Any nonseparating simple

closed curve of T K can play the role of the characteristic curve and the

surgery dened by such a curve is sometimes called a rational surgery The

surgery formula extends naturally to rational surgeries For surgeries starting

from Zspheres the surgery function F can b e expressed only in terms of linking

numb ers and onevariable AlexanderConway p olynomials See L

There are also some formulae for the Casson invariant of pfold branched

cyclic coverings For a link L in a Zsphere M let R M L b e the pfold cyclic

p

covering of M branched along L obtained from the covering of the exterior of

L asso ciated with the linking numb er with L mo dulo p by lling it in by solid

tori whose meridians are sent to old meridians of L

Prop erty Hoste Ho Let K be a knot in a Zsphere M Let D K be the

untwisted double of K with an clasp then

R M D K pM p K

p

The following Mullins prop erty relates the Walker invariant of fold branched

coverings to the Jones p olynomial V and the oriented signature of links

3 3

Prop erty Mullins Mul Let L be a link in S such that R S L is a

2

Qsphere then

V L L

3

R S L

W 2

V L

3

To prove this formula Mullins studied the variation of R S L

W 2

under a crossing change of L Owing to the fact that the fold branched covering

of the ball of the crossing change is a solid torus such a crossing change induces

3

a surgery on R S L

2

For other pfold cyclic branched coverings a crossing change induces a

handleb o dy replacement This leads us to the following natural question What

can we say ab out A B for a Qsphere obtained by gluing two pieces A



and B along a genus g surface

Our partial answer is the following prop erty of L A Qhand lebody is

a manifold with the same rational homology as a standard handleb o dy For

a manifold A with b oundary the kernel L of the map from H A Q to

A 1

H A Q induced by the inclusion is called the Lagrangian of A

1

Prop erty Let A A B and B be four Qhand lebodies such that A A

B and B are identied via orientationpreserving homeomorphisms with

0 0

and that L L and L L a genus g surface Assume that L L

A B B B A A

fg inside H Q Then

1

A B A B A B A B R A A B B

W  W  W  W 

where R A A B B described below in general is zero if g

Before describing R A A B B in general note that for g and A B

3

B this prop erty is nothing but the additivity of under connected sum The

W

genus one formula when A and B are solid tori is the splicing formula shown

by several authors BN FM for the Casson invariant and generalized by Fujita

to the Walker invariant F In this case starting with A B there is a unique



way of lling in A with a solid torus B having the right Lagrangian A B

and A B are similarly welldetermined and the Walker invariant of the lens

space A B is a known Dedekind sum

Now let us decrib e R A A B B under the hyp otheses of Prop erty

0

from H A A Q to L which maps the homology The isomorphism

2  A AA

class of a surface S of A A transverse to A to the class of S A



V

3

0

H A A Q to a form I carries the algebraic intersection dened on

2  AA

V

3

0

similarly Let and b e dened on L Dene I

1 g 1 g A BB

two bases for L and L resp ectively that are dual for the intersection form

A B

on Then

 i j  ij

X

0 0

I I R A A B B

AA i j k BB i j k

fijk gf1g g

Remark Let L b e a closed connected surface equipp ed with a ratio

A

nal Lagrangian as ab ove In S D Sullivan proved that any integral form on

V

3

0

for two standard handleb o dies A H Z L may b e realized as a I

1 A AA

and A with b oundary and Lagrangian L

A

A splitting A B of a Qsphere induces the following function on

 AB

the Torelli group of The Torelli group is the group of the isotopy classes of

homeomorphisms of which induce the identity on H For a homeomor

1

phism f of the Torelli group A B denotes the manifold obtained by replacing

f

the underlying identication j B by j f

B B

f A B A B

AB W f W 

As a direct corollary of Prop erty we see that g f g f

AB AB AB

is a function of the evaluations of the Johnson homomorphism at f and g see J

Second denition p for a denition of the Johnson homomorphism which is

V

3

1

H With completely dier a homomorphism from the Torelli group to

ent metho ds based mainly on Johnsons study of the Torelli group S Morita

proved this corollary for Heegaard splittings of Zspheres Mo Theorem

but he did not think that it extended to general emb eddings Mo Remark

The ab ove corollary also proves that when A B is a Zsphere the function

induced by the Rohlin invariant denes a homomorphism from the

AB

8

Torelli group to ZZ These homomorphisms were rst studied by J Birman

and R Craggs BC they are the socalled BirmanCraggs homomorphisms

It is worth mentioning that the b est natural generalization of Prop erty

that may b e exp ected for the generalized Casson invariant of Section is true

L This generalized Casson invariant also admits a homogeneous denition

via Kontsevich integrals LMMO Both of these prop erties together with the

homogeneous surgery formula enhance the naturality of the generalization of

prop osed in Section

To prove Prop erty we rst nd a sequence of simple surgeries on links

transforming A into A and staying among the Qhandleb o dies with Lagrangian

L Then we apply the surgery formula of L BL to these surgeries and we

A

analyse how the involved formulae dep end on B when B varies among the Q

handleb o dies with b oundary A and with xed Lagrangian

This analysis led us L to construct a tautological generalization of Alexan

der p olynomials to manifolds with b oundary which may b e useful to prove

other prop erties of the Casson invariant We conclude this article with a brief

presentation of this function called the Alexander function which will allow us

to dene the normalized several variable Alexander p olynomial

More ab out Alexander p olynomials the Alexan

der function

All the assertions of this section are proved in L Section Here A denotes

a connected manifold with nonempty b oundary and with nonnegative genus

g g A A denotes the group ring

A

H A

1

Z

A

TorsionH A

1

L

Recall that Z expx as a Zmo dule that its multipli

H A

A

x

Torsion

cation sends expx expy to expx y and that the units of are its

A

elements of the form expx H ATorsion

1

The maximal free ab elian covering of A is denoted by A and the cov

ering map from A to A by p We x a basep oint in A The mo dule

A A

1

H A p Z is denoted by H

1 A

A

Denition The Alexander function A of A is the morphism

A A

g

A H

A A A

which is dened up to a global multiplication by a unit of as follows Take

A

a presentation of H over with r g generators and r relators

A A 1 r +g

which are linear combinations of the Let u u u

1 r A i 1 g

V

g

H Then A u is dened by the equality b e an element of

A A

A u u

A

where the u are represented as combinations

1 r 1 r +g i

V L

r +g r +g

of the and the exterior pro ducts are to b e taken in

j A i

i=1

Of course A u is just the order of the mo dule H u But

A A A A i

hop efully some of the prop erties of A mentioned b elow will convince the reader

A

that it may b e interesting to work with a xed normalization of A

A

Fix a preferred lift of in A Let denote the b oundary map from H

0 A

1

Once a normalization of A is xed A satises to H p

A 0 A A A 0

A

the easy prop erty

g

For any v v v H for any u H

1 g A

A

g

X

u

v A v A v u

i A A

v

i

i=1

u

v v u v v where v v v and v

1 i1 i+1 g 1 g

v

i

This prop erty shows that the next prop erty of the Alexander function gives

a consistent denition of the Reidemeister torsion which yields the Alexander

p olynomial If A is a link exterior then for any element u of H

A

A u u A

A

If A is a several comp onent link exterior then A b elongs to it is dened

A

up to a multiplication by a unit of

A

1

In fact a wellchosen multiplication by an element of the form exp x

2

A where the conjugation makes the Reidemeister torsion satisfy A

sends expx to expx Mi Thus the Reidemeister torsion is an element

1

dened up to sign in Z H ATorsion ZH A Q The choice of an ori

1 1

2

entation O of the vector space H A R H A R suppresses the sign indeter

1 2

mination and allows one to dene A O ZH A Q unambiguously See

1

T L If A is the exterior of an ncomp onent link L in a Qsphere M such

an orientation O is unambiguously dened by a basis of H A R H A R

L 1 2

of the form m m T K T K where m and T K denote

1 n 1 n1 i i

th

the oriented meridian and the b oundary of the tubular neighb orho o d of the i

comp onent of L resp ectively

Note that for a general A a basis M fm m g of H A Q induces

1 n 1

the natural ring inclusion from ZH A Q into the ring Qx x of

M 1 1 n

formal series in the x expm expx

i M i i

In particular if A is the exterior of a several comp onent link L in a Q

sphere M then we use the natural basis M of the meridians of L to dene the

Alexander series

D L M n L O

T

M L

which is equivalent to the several variable Alexander p olynomial

D L

n1

L t expx t expx

1 1 n n

jH M j

1

In general a morphism allows us to dene the order of an element

M

of as the order of its image under It do es not dep end on M Sim

A M

ilarly we will sp eak of the low degree parts of the elements of Indeed

A

the information required to compute the co ecients of the surgery formula

Prop erty is contained in the lowdegree parts of Alexander functions im

ages Thus it is worth noting that the degree part of A u is A u

A A

jH AZp u j and that the order of A u is greater or equal than the

1 A i A

dimension of H A QQp u The Alexander function also satises the

1 A i

following interesting prop erty which relates the low degree parts of some of its

images to algebraic intersections

Prop osition For any A m where A is a Qhand lebody whose boundary

is equipped with two systems of curves and m m m

1 g 1 g

as in Figure such that the homology classes of the generate L

i A

g

X

j

I expm O A m A m

A i j k i A A

m

k

i=1

where O makes up for an element of of order greater or equal than and

A

is the standard hand lebody with boundary A where the bound disks

i

g

2 1

m

m m g

1 2

Figure Two systems of curves on A

Recall that I is dened in Section Though denotes the curve its

i i

homology class and the class of a lifting of the curve joined to the basep oint

i

in H dep ending on the context the statement is unambiguous

A

It is also worth observing the natural go o d b ehaviour of Alexander func

tions under the two op erations Adding a handle to A Performing a

connected sum along the b oundary of two manifolds A and B A lot of prop er

ties of Alexander p olynomials can b e derived from this natural b ehaviour More

generally if A is a submanifold of the interior of a manifold B in order to

B n A A and the inclusion from A into compute A it is enough to know

A B

B n A

Let us use these remarks to b e more sp ecic ab out the sign determination

of the Alexander series

Let L K b e a link in a Qsphere M g Consider a

i

if1g g

regular neighb orho o d of a graph made of the K and paths joining them to the

i

basep oint This is a handleb o dy which is a connected sum along b oundaries

of the T K Removing the interior of this handleb o dy from M yields a Q

i

handleb o dy A whose b oundary is equipp ed with the meridians m and some

i

longitudes of the K which sit there as in Figure We let denote the

i i i

b oundary of the genus one subsurface of A with connected b oundary containing

m and

i i

1

Then up to units of the form expx H ATorsion for any j k

1

2

f g g

m

j

A

A

k

D L sig nA m

A L

m

j

Now the denition of the co ecient is complete and we know enough

ab out the surgery formula Thus we can apply it together with the helpful

formalism intro duced ab ove and we are hop efully ready to discover more prop

erties for the Casson invariant

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Christine Lescop CNRS UMR

Institut Fourier BP

SaintMartindHeres cedex

FRANCE

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