SURGERY EQUIVALENCE AND FINITE TYPE INVARIANTS FOR HOMOLOGY 3-SPHERES L. FUNAR Abstract. One considers two equivalence relations on 3-manifolds relate

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1 SURGERY EQUIVALENCE AND FINITE TYPE INVARIANTS FOR HOMOLOGY 3-SPHERES L. FUNAR Abstract. One considers two equivalence relations on 3-manifolds related to nite type invariants. The rst one requires to have matching invariants in degree less than k + 1 and it is based on a ltration introduced by Garoufalidis and Levine ([4, 1]). The other one allows manifolds to be cut open along embedded surfaces and twist by an element of the k + 1-th term of the lower central series of the group of BSCC maps. The main result states the two relations coincide on the level of homology 3-spheres. The analogous result for the Torelli group was announced by Habiro ([6]), using claspers theory. Similar results for knots and Vassiliev invariants were obtained by Gusarov, Habiro and Stanford ([10]). 1. Finite type invariants and H 1 -bordism classes 1.1. The subgroup of BSCC maps. Let g (respectively g;1 ) be a closed oriented surface of genus g 2 (with one hole). The mapping class group M g;1 is the group of orientation preserving dieomorphisms of the surface g;1 modulo isotopies. Denote by F(2g) the fundamental group 1 ( g;1 ), which is a free group of rank 2g, and by H the homology group H 1 ( g;1 ; Z). There is a canonical surjective morphism onto the symplectic group M g;1?! Sp(H), whose kernel is, by denition, the Torelli group I g;1. It is well known that M g;1 is generated by Dehn twists. An innite set of generators of I g;1 was given by Powell: a) twists around bounding simple closed curves (BSCC-maps), b) opposite twists on a (bounding) pair of disjoint homologous simple closed curves, each of which is nonbounding (BP-maps). Johnson (see [7]) rened Powell's result and showed that only BP-maps bounding a genus 1 subsurface are sucient to generate I g;1 for g 3. Notice that for g = 2 all BP-maps are trivial. We denote by K g;1 I g;1 the subgroup generated by BSCC-maps. One can dene in the same manner the corresponding subgroups I g and K g of the mapping class group M g of a closed surface g. Notice that K g is a normal subgroup of I g of innite index when g 3, which is equal to I 2 when g = 2. The groups I g;1 and I g are nitely generated for g 3, but I 2 is free on innitely many generators. The group K g;1 is the kernel of the rst Johnson homomorphism 1 : I g;1! V3 H, or equivalently the kernel of the natural morphism M g;1! Aut (F(2g)=[F(2g); [F(2g); F(2g)]]). Notice that the groups K g;1 nilpotent since I g;1 are residually torsion free nilpotent (see e.g. [8]). are residually torsion free 1.2. H 1 -bordism classes. Two (closed oriented) 3-manifolds are H 1 -bordant (see [2]) if there exists an oriented cobordism between them which is a product on H 1 (?; Z). Moreover it is proved in [2] that: Date: July 28, Mathematics Subject Classication. Primary 57 N 10, Secondary 57 R 20. Partially supported by a Canon Foundation grant. This paper is available electronically at 1

2 2 L. FUNAR Proposition 1.1. M is H 1 -bordant to M 0 i one of the following equivalent conditions is satised: 1. There exists an isomorphism H 1 (M; A)! H 1 (M; A) which induces isomorphisms on the linking pairing and on the triple cup products with arbitrary coecient A. 2. There exists a Heegaard splitting M = A + g [A? g and ' 2 K g such that M 0 is obtained by twisting the gluing map of the two handlebodies by ', i.e. M 0 = A + g [ ' A? g Finite type invariants. Let us denote by M (respectively HS) the set of orientation preserving dieomorphism classes of 3-manifolds (respectively integer homology 3-spheres) and by ZM (respectively ZHS) the free Abelian Z-module spanned by M (respectively HS). A connected, oriented and separating surface i : g,! M embedded in the 3-manifold M is called an admissible surface in M. For any ' 2 K g, one denotes by M(i; ') the 3-manifold obtained by cutting open M along i( g ), twisting by ' and gluing back. The map sending ' into M(i; ') extends is the by linearity to a homomorphism ZK g! ZM. We restrict this map to IK m g! ZM where I Kg augmentation ideal of ZK g. We introduce (after [4]) the a ltration on ZM as follows: Denition 1.1. Denote by F K m(zm) the submodule of ZM spanned by the images of I m K g;1 over all admissible surfaces g,! M, and all 3-manifolds M. We could set also F HK m i( g ) are allowed. In [4] it is proved that F HK m! ZM, (ZM) for the submodule arising when only Heegaard admissible surfaces (ZHS) = F K m(zhs) for homology spheres. Following [3], one can also characterize this ltration by using boundary links. A link L M is called a boundary link if each component bounds an oriented Seifert surface such that these Seifert surface are all disjoint from each other. For any framed link (L; f) in M one sets [M; L; f] = X L0L(?1) jl0j M L 0 ;f 0 2 ZM where the sum is over all sublinks of L, f 0 is the restriction of the framing f and jl 0 j is the number of components of L 0. Denote by F B m(zm) the submodule of ZM spanned by [M; L; f] for all m- components unit-framed boundary links (L; f). Then the equality F K (ZM) = F B (ZM) holds (the proof of [4] extends to general 3-manifolds). Recall that a map v : M! A taking values in an Abelian group A is an invariant of degree m if it satises vj F K m+1 (ZM) = 0 (when properly extended by linearity). The manifolds M and N are (Vassiliev or nite type or simply) k-equivalent if they are not distinguished by nite type invariants of degrees k. This is the same as asking for M? N 2 F K k+1 (ZM). Remark 1.1. The degree 0 invariants correspond to the H 1 -bordims classes of 3-manifolds. Hence it is natural to study nite type invariants separately, for each family M(M) containing the manifolds which are H 1 -bordant to M. This point of view is used in [1] in studying the Ohtsuki ltration. In particular the class of homology spheres deserves a special attention in the sequel. Notice that the inclusion F HK (ZM(M)) F K (ZM(M)) might be strict, when M is not a HS Surgery k-equivalence based on the series K. Consider another ltration, coming from the k-surgery equivalence (or surgery k-equivalence) based on the series of groups K. Denition 1.2. Two manifolds are (Heegaard) k-surgery equivalent if N = M(i; '), for some (Heegaard) surface i( g ) M and some ' 2 LCS k+1 (K g ). Here and henceforth LCS k (G) is the k-th term of the lower central series of the group G. The ltration Sm(ZM) K (respectively S HK m (ZM)) is the span of the dierences M? N of (Heegaard) k-surgery equivalent manifolds M and N.

3 SURGERY EQUIVALENCE AND FINITE TYPE INVARIANTS FOR HOMOLOGY 3-SPHERES 3 Notice that the k-surgery equivalence is the same as Heegaard k-surgery equivalence for HS, thus Sm(ZHS) K = S HK m (ZHS). In fact any surface in a HS can be made a Heegaard surface by adding 1-handles. The gluing map for the new decomposition is the connect sum of the former one with the identity and it lies therefore in the same term of the corresponding lower central series Statement of the result. Theorem 1.1. Two HS are k-equivalent if and only if they are surgery k-equivalent. Remark 1.2. A similar result holds (see [6]) for the two equivalence relations related to the Torelli group (one replaces K g by I g ). The counterpart ltration is the so-called blink ltration from [4], which is equivalent to Ohtsuki's ltration or A-type ltration from [6] up to renumbering. The proof which will be given below applies with only minor modications. The present result is slightly more precise since [K g;1 ; K g;1 ] is a proper subgroup of I g;1, whereas in the comparison of the ltrations we replace 2m iterated commutators from I g;1 by m iterated commutators of K g;1. The similar question about the equality of ltrations over Q instead of Z (this amounts to replace LCS k (K g;1 ) or LCS k (I p g;1 ) by its rational closure - or radical - LCS k (K g;1 ) = fg 2 K g;1 ; 9n; g n 2 LCS k (K g;1 )g) was asked in [4]. Notice that in the case of the Torelli group the graded quotients of the lower central series might have torsion (in contrast with the series of rational closures which is the most rapidly descending p series). The simplest example is H 1 (I g;1 ; Z) LCS 2 (I g;1 ) which contains the nontrivial 2-torsion group, described explicitly by Johnson and LCS 2 (I g;1 ) arising from the Birman-Craggs homomorphisms I g;1! Z=2. Notice that p LCS 2 (I g;1 ) = K g;1. Thus the possible relationship between the graded algebra of F K and the Malcev completion of Torelli groups would rather come from the rational closure series of groups. It is proved in [4] that F K \ m(zhs) = F3m(ZHS), as where F3m(ZHS) as is Ohtuski's ltration (de- ned as above, but using all unit-framed algebraically-split links) and we used the notation \ F T 1 m (ZHS) = F n=0 m (ZHS) + F3n(ZHS). as Gusarov ([5]) announced the equality Fm(ZHS) K Q = F3m(ZHS) as Q. Thus the associated graded rings over Q are isomorphic. The proof we give below follows that of Stanford ([10]) for the knot case, but it is simpler in many respects and avoids the use of the (very interesting) Habiro claspers theory. We are thankful to C.Blanchet, T.Kitano, C.Lescop for useful suggestions and comments The plan of the proof. The \if" part of the theorem 1.1 is obvious. If I G ZG denotes the augmentation ideal for the group ring of the group G, then its dimension subgroup D k (G) = (1 + I k G) \ G contains the corresponding lower central series group LCS k (G). This shows that S K (ZM) F K (ZM). The theorem thus follows from: Proposition 1.2. The k-surgery equivalence is an equivalence relation. Proposition 1.3. The set of HS modulo k-surgery equivalence is a group (under the connected sum of manifolds). Proposition 1.4. Two k-equivalent HS are k-surgery equivalent.

4 4 L. FUNAR 2. Proofs Before proceeding to the proofs we will need some preparations. We will use throughout this section some argument concerning stabilizations of surfaces and mapping classes. It is well-known that injective homomorphisms M g! M g+1 or M g! M g;1 do not exist, in general. We will work then, instead of M g with the mapping class groups M g;1 of g;1. This time we can use the natural injective morphisms M g;1! M g+1;1, and make connected sum of mapping classes. We could dene the k-surgery equivalence and the ltration F K in the same way, but using 1-holed surfaces. The manifolds M and N are equivalent if N = M(i; ') for some admissible embedding i, such that ' is the image in K g of an element of LCS k+1 (K g;1 ). The ltration is given now by the images of IK m g;1 into ZM. However the new dened ltrations coincide with the former ones because the maps LCS k (K g;1 )! LCS k (K g ) and I m K g;1! I m K g are surjective. The possibility to work with mapping classes of 1-holed surfaces will be freely used throughout the arguments below Proof of Proposition 1.3. We have to check that the k-surgery equivalence is transitive. Consider then N = M(i; ') and N 0 = M(i 0 ; ' 0 ). Proposition 1.1 implies that we can choose both i and i 0 to be Heegaard embeddings. Furthermore two Heegaard surfaces are stably homeomorphic. Notice that using a stabilization we change the gluing map ' into ']id, the connect sum with identity on the boundary of the stabilizing handle. The gluing map is still an element of the corresponding K g, where g is the appropriate genus. We can therefore suppose that the Heegaard splittings are homeomorphic. Then N 0 can be obtained from N by cutting open along the Heegaard surface and inserting the twist by ' 0 '? Proof of Proposition 1.4. Denote by S;k the k-surgery equivalence. S;k Lemma 2.1. If M 1 M 0 S;k 1 and respectively M 2 M 0 S;k 2 then M 1 ]M 2 M1]M 0 2. Proof. If M 0 1 = M 1 (i 1 ; ' 1 ) and M 0 2 = M 2 (i 2 ; ' 2 ), then we can use the connected sum Heegaard embedding i 1 ]i 2 into M 1 ]M 2. Then M 0 1]M 0 2 = (M 1 ]M 2 )(i 1 ]i 2 ; (' 1 ]id)(id]' 2 )). When stabilizing with identity we remain in the same term of the lower central series, hence the claim follows. It remains to show that the Abelian monoid structure induced by ] is a group structure: Lemma 2.2. Every equivalence class in HS= S;k has an inverse. Proof. It suces to prove that for a k-surgery trivial homology sphere M there exists some HS M 0 such that M]M 0 is k + 1-surgery trivial. Since any HS is 0-surgery trivial an inductive use of this claim proves the lemma. Set S 3 for the 3-sphere. Consider M = S 3 (i g ; ') such that S 3 (i g ; ') is homeomorphic to S for some 2 LCS k+1 (K g;1 ), where i g : g S 3 is the (standard, unique) Heegaard embedding of a genus g surface into the sphere. We prove that M 0 = S 3 (i g ; ) veries the claim. More generally, for any ' 2 K g;1 and 2 LCS k+1 (K g;1 ), S 3 (i g ; ')]S 3 (i g ; ) is k + 1-surgery equivalent to S 3 (i g ; ' ). Let consider the Heegaard embedding i 2g of genus 2g. Then S 3 (i g ; ')]S 3 (i g ; ) is homeomorphic to S 3 (i 2g ; (']id)(id] )). If is the homeomorphism of 2g which interchanges the two halves g;1 ) of 2g then we can express id] =?1 ( ]id). Hence the connect sum is the same as S 3 (i 2g ; (' ]id)[(?1 ]id;?1 ]), which is k + 1-surgery equivalent to S 3 (i 2g ; (' ]id)) = S 3 (i g ; ' ), because [(?1 ]id;?1 ] 2 LCS k+2 (K 2g;1 ).

5 SURGERY EQUIVALENCE AND FINITE TYPE INVARIANTS FOR HOMOLOGY 3-SPHERES Proof of Proposition 1.5. Set k for the k-equivalence relation. Gusarov and Habiro (see [5, 6]) proved that HS k is an Abelian group under the connected sum. We have two composition maps HS ZHS! Z S;k! HS S;k ; HS ZHS! Z k! HS k : where the right hand side arrows are the Z-linear extension of the identity map between the two groups. In order to identify these two maps it is sucient to prove the kernel of the second one is contained in the kernel of the former. Denote by M(i; P a m ' m ) = P a m M(i; ' m ) 2 ZM, for scalars a m and mapping classes ' m. Denition 2.1. A relator of order n and length m is an element of ZHS having the form S 3 (i; (' 1? 1)(' 2? 1):::(' m? 1) ) where n i is the greater natural such that ' i 2 LCS ni (K g;1 ), and n = P i n i, 2 K g. This makes sense because K g;1 is residually nilpotent for g 3. From the denition Fm(ZHS) K is generated by relators of length m and order m. In fact any other element of the form M(i; (' 1? 1)(' 2? 1):::(' m? 1) ) can be transformed into a relator. In fact if i induces a Heegaard splitting of M with gluing map 2 K g, then the previous h relation i is the same as S 3 S;k (i g ; (' 1? 1)(' 2? 1):::(' m? 1) ). Furthermore the kernel of ZHS! Z HS= is generated by relators of length 1 and order k. The kernel of the second projection Z[HS= S;k HS= S;k ]! is generated by the composite combinations (which we call composite relators) of the form M 1 ]M 2? M 1? M 2. Consequently the elements of the form S 3 (i; ' 1? 1)]S 3 (i; ' 2? 1) are also sums of composite relators. The theorem is a consequence of the following: Proposition 2.1. Any relator of order n is a linear combination (over Z) of relators of length 1 and order n and of composite relators. Proof. Set C n be the span of all relators length 1 and order n and of composite relators. Suppose on the contrary that there is one relator of order n which does not belong to C n. Choose one such of minimal length m, and among those with minimal length, one with maximal order. Such one should exist because, when the order is large enough the relator should be in C n. In fact, there exists m such that ' m 2 LCS n (K g;1 ) (if the order is large enough) and the relator can be rewritten as a sum of elements of the form as S 3 (i; w(' m? 1) ) = S 3 (i; (w' m w?1? 1)w ), and w' i w?1 2 LCS n (K g;1 ) hence all these are relators of length 1 and order n. Assume now that the contradicting relator is S 3 (i g ; (' 1? 1):::(' m? 1) ), ' j ; 2 K g;1 K 2g;1. This is equivalent to S 3 (i 2g ; ' 1? 1):::(' m? 1) ), where i 2g is the Heegaard embedding of genus 2g and by abuse of notation we identify x 2 K with x]id 2 K 2g;1. We can replace our relator by S 3 (i 2g ; ' 1? 1)(' 2? 1):::(' m?1? 1)(?1 ' m? 1) ). In fact the difference between the former and this one is S 3 (i 2g ; (' 1? 1)(' 2? 1):::(' m?1? 1)([?1 ; ' m ]? 1)' m ) which has higher order since [?1 ; ' m ] 2 LCS nm+1 (K 2g;1. Similarly we can interchange y and x 0 m? 1 =?1 ' m? 1 modulo relators of greater order (and the same length) and obtain S 3? i 2g ; (' 1? 1)(' 2? 1):::(' m?1? 1) (?1 ' m? 1) = = S 3 (i 2g ; (' 1? 1)(' 2? 1):::(' m?1? 1) ) ]S (i 2g ; ' m? 1)) :

6 6 L. FUNAR The last relator is a linear combination (with integer coecients) of composite relators, and we are done. References [1] T. Cochran and P. Melvin, Finite type invariants of 3-manifolds, math.gt/ [2] T. Cochran, A. Gerges and K. Orr, Surgery equivalence relations on 3-manifolds, preprint, [3] S. Garoufalidis, On nite type 3-manifold invariants I, J. Knot Theory Ramications, 5 (1996), 441{461. [4] S. Garoufalidis and J. Levine, Finite type 3-manifold invariants, the mapping class group and blinks, J. Di. Geometry, 47 (1997), 257{320. [5] M. Gusarov. Finite type invariants and n-equivalence of 3-manifolds. preprint, [6] K. Habiro, Claspers and nite type invariants of links, preprint, [7] D. Johnson, A survey of the Torelli group, Low-dimensional topology (San Francisco, Calif., 1981), Contemp. Math., v.20, 1983, [8] R. Hain, Innitesimal presentations of the Torelli groups, J. Amer. Math. Soc., 10(1997), 597{651. [9] T. Ohtsuki, Finite type invariants of integral homology 3-spheres, J. Knot Theory Ramications, 5 (1996), 101{115 [10] T. Stanford, Vassiliev invariants and knots modulo pure braid subgroups, mathgt/ Institut Fourier BP 74, University of Grenoble I, Saint-Martin-d'Heres cedex, France address: funar@fourier.ujf-grenoble.fr