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On the Casson Invariant 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 link 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
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