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CHARACTERIZATION OF Y -EQUIVALENCE 2 FOR HOMOLOGY CYLINDERS 3 0 0 GWEŃAE¨LMASSUYEAUandJEAN-BAPTISTEMEILHAN 2 Laboratoire Jean Leray, UMR 6629 CNRS/Université de Nantes 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 03, France n a [email protected], [email protected] J 6 ABSTRACT ] For Σ a compact connected oriented surface, we consider homology cylinders over T Σ: thesearehomologycobordismswithanextrahomologicaltrivialitycondition. When G considereduptoY2-equivalence,whichisasurgeryequivalencerelationarisingfromthe . Goussarov-Habirotheory,homologycylindersformanAbeliangroup. h Inthispaper,whenΣhasoneorzeroboundarycomponent,wedefineasurgerymapfrom t a acertainspaceofgraphstothisgroup. Thismapisshowntobeanisomorphism,with m inversegivenbysomeextensionsofthefirstJohnsonhomomorphismandBirman-Craggs homomorphisms. [ Keywords: homologycylinder,finitetypeinvariant,clover,clasper. 2 v Date: March18,2002and,inrevisedform,September 5,2002 9 7 1. Introduction 1 3 0 1.1. Homology cylinders 2 Homologycylindersareimportantobjects inthe theoryoffinite type invariants 0 / ofGoussarov-Habiro: they havethus appearedinboth [6] and[4]. Letus recallthe h definition of these objects. t a LetΣbeacompactconnectedorientedsurface. Ahomology cobordism overΣis m a triple (M,i+,i−) where M is a compact oriented 3-manifold and i± :Σ - M : are oriented embeddings with images Σ±, such that: v i X (i) i± are homology isomorphisms; r (ii) ∂M =Σ+∪(−Σ−) and Σ+∩(−Σ−)=±∂Σ±; a (iii) i+| =i−| . ∂Σ ∂Σ Homologycobordismsareconsidereduptoorientation-preservingdiffeomorphisms. When (i−)−1 ◦ (i+) : H (Σ;Z) - H (Σ;Z) is the identity, M is said to be ∗ ∗ 1 1 a homology cylinder. The set of homology cobordisms is denoted here by C(Σ), and HC(Σ) denotes the subset of homology cylinders. If M = (M,i+,i−) and 2 N =(N,j+,j−) are homology cobordisms, we can define their stacking product by M ·N :=(M ∪i−◦(j+)−1 N,i+,j−). This product induces a monoid structure on C(Σ), with HC(Σ) a submonoid. The unit element is 1 := (Σ×I,Id,Id), where I is the unit interval [0,1] and where Σ a collar of Σ± is stretched along ∂Σ×I so that the second defining condition for homology cobordisms is satisfied. Habiroin[6, §8.5]outlinedhowhomologycylinderscanserveas apowerfultool in studying the mapping class groups of surfaces (see [3], [5], [12]). The connection lies on the homomorphism of monoids C- T(Σ) HC(Σ) sendingeachhintheTorelligroupofΣtothemappingcylinderC =(Σ×I,Id,h) h (with, as above, a collar of Σ± stretched along ∂Σ×I). In the sequel, we restrict ourselves to the following two cases: (i) Σ = Σ is the standard closed oriented surface of genus g ≥ 0, which here g is referred to as the closed case; (ii) Σ=Σ is the standard compact oriented surface of genus g ≥0 with one g,1 boundary component, which here is referred to as the boundary case. The usual notations T = T(Σ ) and T = T(Σ ) for the Torelli groups will be g,1 g,1 g g used. Also denote by H the first homology group of Σ with integer coefficients, by • the intersection form on H and by (x ,y )g a symplectic basis for (H,•). i i i=1 1.2. Y -equivalence k The theory of finite type invariants of Goussarov-Habiro has come equipped with a topological calculus toolbox: this was called calculus of claspers in [6] or alternatively clovers in [2]. We will assume a certain familiarity of the reader with these techniques. Inparticular,letusrecallthat,fork ≥1aninteger,theY -equivalence1istheequiv- k alence relation generated by surgery on connected clovers of degree k. Following Habiro in [6], we can then define a descending filtration of monoids C(Σ)⊃C (Σ)⊃C (Σ)⊃···⊃C (Σ)⊃··· 1 2 k where C (Σ) is the submonoid consisting of the homology cobordisms which are k Y -equivalentto the trivialcobordism1 . Note the following fact, a proofofwhich k Σ has been inserted in §4. Proposition 1.1. If Σ=Σ or Σ , then HC(Σ)=C (Σ). g g,1 1 As mentioned by Habiro, we can show from the calculus of clovers that for every k ≥1, the quotient monoid C (Σ):=C (Σ)/Y k k k+1 isanAbeliangroup. Inparticular,C (Σ)istheAbeliangroupofhomologycylinders 1 over Σ up to Y -equivalence. This group is the subject of the present paper. 2 1Thisequivalencerelationiscalled(k−1)-equivalence in[4],andAk-equivalencein[6]. 3 For k ≥ 2, Habiro gives a combinatorial upper bound for the Abelian group C (Σ). Precisely, he defines A (H) to be the Abelian group (finitely) generated k k by unitrivalent graphs of internaldegreek, with cyclic orientationat eachtrivalent vertex and whose univalent vertices are labelled by elements of H and are totally ordered. These graphs are considered modulo the well-known AS, IHX, multilinearity relations, and up to some “STU-like relations” dealing with the order of the univalent vertices. In the closed case, some relations of a symplectic type can be added. Then, there is a surjective surgery map A (H) ψ--k C (Σ) k k sending each graph G to (1 ) , where G˜ is a clover in the manifold 1 with G as Σ G˜ Σ associated abstract graph, whose leaves are stacked from the upper surface Σ×1 according to the total order, framed along this surface and embedded according to the labels of the corresponding univalent vertices. The fact that ψ is well-defined k also follows from the calculus of clovers. Asforthecasek =1,Habirodoesnotdefineanyspaceofgraphsbutannounces the following isomorphisms C (Σ )≃Λ3H ⊕Λ2H ⊕H ⊕Z (1.1) 1 g,1 (2) (2) 2 C (Σ )≃Λ3H/(ω∧H)⊕Λ2H /ω ⊕H ⊕Z (cid:26) 1 g (2) (2) (2) 2 where H =H ⊗Z and where (2) 2 g ω = x ∧y ∈Λ2H i i i=1 X is the symplectic element. This fact has been used afterwards in [12]. The goal of this paper is to prove these isomorphisms, in a diagrammatic way, by again defining a surgery map A (P) ψ-1 C (Σ). 1 1 The space of graphs A (P) and the map ψ appear to be meaningfully different 1 1 from A (H) and ψ for k > 1, making thus the case k = 1 exceptional. Indeed, k k their definition will involve both the homology group H and Spin(Σ), the set of spin structures on Σ. 1.3. The Abelianized Torelli group - We denote by Ω the set of quadratic forms with • : H ×H Z as g (2) (2) 2 associated bilinear form, namely q- Ω = H Z : ∀x,y ∈H , q(x+y)−q(x)−q(y)=x•y . g (2) 2 (2) n o Note that Ω is an affine space over H , with action given by g (2) ∀q ∈Ω ,∀x∈H , x·q :=q+x•(−). g (2) - Thus, among the maps Ω Z , there are the affine functions, and more gen- g 2 erally there are the Boolean polynomials which are defined to be sums of products 4 of affine ones (see [8, §4]). These polynomials form a Z -algebra denoted by B , 2 g which is filtered by the degree (defined in the obvious way): B(0) ⊂B(1) ⊂···⊂B . g g g For instance, B(1) is the space of affine functions on Ω ; the constant function g g - 1 : Ω Z sending each q to 1 and, for h ∈ H, the function h sending each q g 2 to q(h) are affine functions. Note the following identity: (1.2) ∀h ,h ∈H, h +h =h +h +(h •h )·1∈B(1). 1 2 1 2 1 2 1 2 g Another example of Boolean polynomial is the quadratic Boolean function g α= x ·y , i i i=1 X which is known as the Arf invariant. For any basis (e )2g for H, there is an i i=1 isomorphism of algebras: Z [t ,...,t ] 2 1 2g (1.3) B ≃ g t2 =t i i sending 1 to 1 and e to t . i i Recall now from [8], that the many Birman-Craggs homomorphisms can be summed up into a single homomorphism β β B(3) T - B(3) or T - g , g,1 g g α·B(1) g accordingtowhetheroneisconsideringtheboundarycaseortheclosedcase. Recall also from [9] that the first Johnson homomorphism is a homomorphism η η Λ3H T 1- Λ3H or T -1 . g,1 g ω∧H Form the following pull-back: Λ3H × B(3) - B(3) Λ3H(2) g g q ? ? Λ3H - Λ3H , (2) −⊗Z 2 where the map q is the canonical projection B(3) - B(3)/B(2) followed by the g g g isomorphismB(3)/B(2) ≃Λ3H whichidentifiesthecubicpolynomialh h h with g g (2) 1 2 3 h ∧h ∧h (this is well-defined because of (1.2) and (1.3)). 1 2 3 We denote by S the subgroupof this pull-back correspondingto ω∧H ⊂Λ3H and α·B(1) ⊂ B(3). Johnson has shown in [10] that, under the assumption g ≥ 3, the g g homomorphisms η and β induce isomorphisms 1 Tg,1 (η1,β-) Λ3H × B(3) and Tg (η1,β-) Λ3H ×Λ3H(2) Bg(3). T′ ≃ Λ3H(2) g T′ ≃ S g,1 g 5 Remark 1.2. Notethat,becauseof(1.3),thecodomainsofthesemapsarerespec- tivelynon-canonicallyisomorphictoΛ3H⊕Λ2H ⊕H ⊕Z andΛ3H/(ω∧H)⊕ (2) (2) 2 Λ2H /ω ⊕H ⊕Z . (2) (2) (2) 2 1.4. Statement of the results In §2, we will construct the space of graphs A (P) and the surgery map ψ : 1 1 - A (P) C (Σ). Spin structures play a prominent role in their definitions. 1 1 Observe that, C (Σ) being an Abelian group, the mapping cylinder construction 1 induces a group homomorphism T(Σ) C - C (Σ). T(Σ)′ 1 AspointedoutbyGaroufalidisandLevinein[3]and[12],Johnsonhomomorphisms - and Birman-Craggs homomorphisms factor through C :T(Σ) HC(Σ). These extensions will be detailed in §3. Next, we will specify in §4 an isomorphism ρ : A (P) - Λ3H × B(3) and 1 Λ3H(2) g the following two theorems will be proved from the previous material. Theorem 1.3. In the boundary case, the diagram ψ C T 1 - (cid:27) g,1 A (P) C (Σ ) 1 1 g,1 T′ g,1 @ (cid:0) @ (cid:0) @ (η1,β) (cid:0) ρ @ (cid:0) (η1,β) @R ? (cid:9)(cid:0) Λ3H × B(3) Λ3H(2) g commutes and all of its arrows are isomorphisms, except for the two maps starting from T /T′ when g <3. g,1 g,1 Theorem 1.4. In the closed case, the diagram A (P) ψ C T 1 1 - (cid:27) g C (Σ ) ρ−1(S) 1 g T′ g @ (cid:0) @ (cid:0) @ (η1,β) (cid:0) ρ @ ? (cid:0) (η1,β) @RΛ3H × B(3)(cid:9)(cid:0) Λ3H(2) g S commutes and all of its arrows are isomorphisms, except for the two maps starting from T /T′ when g <3. g g Note that Theorem 1.3 and Theorem 1.4 together with Remark 1.2, give Habiro’s isomorphisms (1.1), which are non-canonical. Also, we will easily deduce the following. Corollary 1.5. For Σ=Σ or Σ , let M and M′ be two homology cylinders over g,1 g Σ. Then, the following assertions are equivalent: (a) M and M′ are Y -equivalent; 2 6 (b) M and M′ are not distinguished by degree 1 Goussarov-Habiro finite type invariants; (c) M and M′ are not distinguished by the first Johnson homomorphism nor Birman-Craggs homomorphisms. - Finally, if an embedding Σ ⊂ Σ is fixed, there is an obvious “filling-up” g,1 g - map C (Σ ) C (Σ ), through which the commutative diagrams of Theorem 1 g,1 1 g 1.3 and Theorem 1.4 are compatible. The reader is referred to §4 for a precise statement. 2. Definition of the Surgery Map ψ 1 In this section, we define the space of graphs A (P) and the surgery map ψ 1 1 announced in the introduction. 2.1. Special Abelian groups and the A functor 1 Let us denote by Ab the category of Abelian groups. An Abelian group with special element isapair(G,s)whereGisanAbeliangroupands∈Gisoforderat most2. WedenotebyAb thecategoryofspecialAbeliangroupswhosemorphisms s aregrouphomomorphismspreservingthespecialelements. Wenowdefineafunctor Ab A-1 Ab s inthefollowingway. For(G,s)anobjectinAb ,A (G,s)isthefreeAbeliangroup s 1 generatedbyY-shapedunitrivalentgraphs,whosetrivalentvertexisequippedwith a cyclic orderonthe incident edgesand whose univalent vertices arelabelled by G, subject to some relations. The notation Y[z ,z ,z ] 1 2 3 will stand for the Y-shaped graph whose univalent vertices are colored by z , z 1 2 and z ∈ G in accordance with the cyclic order, so that our notation is invariant 3 under cyclic permutation of the z ’s. The relations are the following ones: i Antisymetry (AS) : Y[z ,z ,z ]=−Y[z ,z ,z ], 1 2 3 2 1 3 Multilinearity of colors : Y[z +z ,z ,z ]=Y[z ,z ,z ]+Y[z ,z ,z ], 0 1 2 3 0 2 3 1 2 3 Slide : Y[z ,z ,z ]=Y[s,z ,z ], 1 1 2 1 2 where z ,z ,z ,z ∈ G. For (G,s) f- (G′,s′) a morphism in Ab , A (f) maps 0 1 2 3 s 1 each generator Y[z ,z ,z ] of A (G,s) to Y[f(z ),f(z ),f(z )]∈A (G′,s′). 1 2 3 1 1 2 3 1 7 - Example 2.6. The map [G (G,0)] makes Ab a (full) subcategory of Ab . It s follows from the definitions that the following diagram commutes: - - Ab Ab s @ @ @ A1 Λ3(−) @ R@ ? Ab. Non-trivial examples will be given in the next paragraph. For future use, note that this category has an obvious pull-back construction extending that of Ab: - (G ,s )× (G ,s ) (G ,s ) 1 1 (G,s) 2 2 2 2 f 2 ? ? - (G ,s ) (G,s) 1 1 f 1 where(G ,s )× (G ,s )isthe subgroupofG ×G consistingofthose (z ,z ) 1 1 (G,s) 2 2 1 2 1 2 such that f (z )=f (z ), and with special element (s ,s ). 1 1 2 2 1 2 2.2. Spin structures and the special Abelian group P In this paragraph, let M be a compact oriented 3-manifold endowed with a Riemannian metric, and let FM be its bundle of oriented orthonormalframes: i p - - -- SO(3) E(FM) M. Lets∈H (E(FM);Z) be the image by i ofthe generatorofH (SO(3);Z)≃Z . 1 ∗ 1 2 Recall that M is spinnable and that Spin(M) can be defined as Spin(M)= y ∈H1(E(FM);Z ), <y,s>=6 0 , 2 which is essentially independ(cid:8)ent of the metric. The manifold M(cid:9)being spinnable, s is not 0 (and so is of order 2). Now, Spin(M) being an affine space over H1(M;Z ) with action given by 2 ∀x∈H1(M;Z ),∀σ ∈Spin(M), x·σ :=σ+p∗(x), 2 we can consider the space A(Spin(M),Z ) 2 of Z -valued affine functions on Spin(M). For instance, 1 ∈ A(Spin(M),Z ) will 2 2 - denote the constant map defined by σ 1. There is a canonical map κ- A(Spin(M),Z ) H (M;Z ). 2 1 2 For f ∈A(Spin(M),Z ), the homology class κ(f) is defined unambiguously by 2 ∀σ,σ′ ∈Spin(M), f(σ′)−f(σ)=<σ′/σ,κ(f)>∈Z , 2 8 where σ′/σ ∈H1(M;Z ) is definedby the affine actionofH1(M;Z ) onSpin(M). 2 2 Another canonical map is e- H (E(FM);Z) A(Spin(M),Z ) 1 2 - sending a x to the map defined by σ < σ,x >. Next lemma gives us a nice understanding of the special Abelian group (H (E(FM);Z),s). 1 Lemma 2.7. a) The following diagram of special groups is a pull-back diagram: e - (H (E(FM);Z),s) A(Spin(M),Z ),1 1 2 (cid:0) (cid:1) p κ ∗ ? ? - (H (M;Z),0) (H (M;Z ),0). 1 1 2 −⊗Z 2 b) Let t be the map t- Oriented framed knots in M H (E(FM);Z) 1 which adds to a(cid:8)ny oriented framed knot K an(cid:9)extra (+1)-twist, and next sends it to the homology class of its lift in FM. Then, (i) t is surjective; (ii) t = t if and only if K and K are cobordant as oriented knots in M K1 K2 1 2 and if their framings with respect to a surface with boundary (K )∪(−K ) 1 2 then differ from each other by an even integer; (iii) if K ♯K denotes the band connected sum of K and K , then t = 1 2 1 2 K1♯K2 t +t ; K1 K2 (iv) the k-framed trivial oriented knot (k ∈Z) is sent by t to k·s. Proof. We begin by proving a). The commutativity of the diagram of special groups is easy to verify. By functoriality, we get a map (H1(E(FM);Z),s) (p∗,-e) (H1(M;Z),0)×(H1(M;Z2),0) A(Spin(M),Z2),1 . The Serresequence associatedto the fibrationFM givesfo(cid:0)r homologywith int(cid:1)eger coefficients: i p - -∗ -∗ - 0 H (SO(3);Z) H (E(FM);Z) H (M;Z) 0. 1 1 1 The bijectivity of (p ,e) follows from the exactness of this sequence. ∗ We now prove b) and we begin with assertion(iv). Let K be a trivial k-framed oriented knot, let ∗∈K and let e=(e ,e ,e )∈p−1(∗) be the framing of K at ∗. 1 2 3 We denote by K˜ the lift of K to FM. Then, as a loop in E(FM), K˜ is homotopic to the loop in the fiber p−1(∗) defined by - [0,1]∋t R (e), 2π(k+1)t where R (with θ ∈ R) denotes the rotation of oriented axis directed by e and θ 3 angle θ. From an appropriate description of the generator of π (SO(3)) ≃ Z , it 1 2 follows that K˜ =(k+1)·s∈H (E(FM);Z), and assertion (iv) then follows. 1 h i 9 Letusmakeanobservation. LetK beanyorientedframedknotinM;sincethe framing of K determines a trivialization of its normal bundle in M, it allows us to restrictanyspin structureonM to K. Recallnowthat the cobordismgroupΩSpin 1 is isomorphic to Z (with generator given by S1 endowed with the spin structure 2 induced by its Lie group structure: see [11, p. 35, 36]). The following observation then makes sense: (2.4) ∀σ ∈Spin(M), e(t )(σ)=(K,σ| ) ∈ΩSpin ≃Z , K K 1 2 and can be derived from an appropriate characterization of the spin structures on the circle (see [11, p. 35, 36]). Let now K and K be some disjoint oriented framed knots in M. There is an 1 2 obviousgenus0surface withboundaryK ♯K ∪˙(−K )∪˙(−K ). Then, accordingto 1 2 1 2 (2.4), we have e(t ) = e(t )+e(t ). Also, p (t )= [K ♯K ]= [K ]+ K1♯K2 K1 K2 ∗ K1♯K2 1 2 1 [K ]= p (t )+p (t ), and so by a), we obtain that assertion (iii) holds for K 2 ∗ K1 ∗ K2 1 and K . 2 We now justify assertion (ii). According to a), t = t if and only if p (t ) = K1 K2 ∗ K1 p (t ) and e(t ) = e(t ). Also, the condition p (t ) = p (t ) holds if and ∗ K2 K1 K2 ∗ K1 ∗ K2 onlyifK andK arehomologousinM. Inthiscase,letS beanembeddedoriented 1 2 surface in M such that ∂S =K ∪˙(−K ). Let k be the framing of K with respect 1 2 i i to S and let K′ be the oriented framed knot obtained from K by adding an extra i i (−k )-twist, so that the framing of K′ is given by S. Then, according to (2.4), i i we have e tK′ = e tK′ . Moreover, applying assertions (iii) and (iv), we obtain: 1 2 e tKi′ = e(cid:0)(tK(cid:1)i)+k(cid:0)i·s.(cid:1)We conclude that e(tK1) = e(tK2) if and only if k1 and k are equal modulo 2, proving thus assertion (ii). 2(cid:0) (cid:1) Let x∈ H (E(FM);Z), then p (x) ∈H (M;Z) can be realized by an oriented 1 ∗ 1 knot K in M: we give it an arbitrary framing. By construction, p (t −x) = 0 ∈ ∗ K H (M;Z), andso by exactnessofthe Serresequence,t −x=ε·swith ε∈{0,1}. 1 K By possibly band-summing K with a trivial (+1)-framed knot when ε = 1, and accordingtoassertion(iii)and(iv),the framedknotK canbe supposedtobe such that t =x; this proves assertion (i). (cid:3) K We now restrict ourselves to the 3-manifold M = 1 = Σ×I where Σ can be Σ Σ or Σ . The inclusion i+ :Σ ⊂ - 1 , with image Σ+, induces an isomorphism g g,1 Σ betweenH andH (M;Z)andabijectionbetweenSpin(Σ)andSpin(M). Asshown 1 byJohnsonin[7],thereisanalgebraicwaytothinkofSpin(Σ). Indeed,thereexists a canonical affine isomorphism ≃- Spin(Σ) Ω , g sendinganyspinstructureσ toaquadraticformq whichcanbedefinedasfollows. σ Letx∈H =H (Σ;Z )berepresentedbyanorientedsimpleclosedcurveonΣ+; (2) 1 2 by framing it along Σ+ and pushing it into the interior of 1 , we get a framed Σ oriented knot K in 1 . Then, Σ (2.5) q (x)=e(t )(σ×I) ∈Z . σ K 2 10 Therefore, according to Lemma 2.7 a), (H (E(F1 );Z),s) is canonically isomor- 1 Σ phic to the special Abelian group defined by the pull-back construction e (H,0)× B(1),1 - B(1),1 (H(2),0) g g (cid:16) (cid:17) (cid:16) (cid:17) p κ ? ? - (H,0) H ,0 (2) −⊗Z 2 (cid:0) (cid:1) whose projections are denoted by p and e, and where κ is the composite ≃ B(1) -- B(1)/B(0) - H . g g g (2) The last isomorphism here identifies h with h for all h ∈ H (this is well-defined (2) by (1.2) and (1.3)). We define the special Abelian group P to be P =(H,0)× B(1),1 , (H(2),0) g (cid:16) (cid:17) and A (P) is the space of graphs announced in the introduction. 1 Remark 2.8. Thus, any element z of P can be written as z = h,h+ε·1 ∈P, with h∈H and ε∈{0,1}. Observe(cid:0)also the fo(cid:1)llowing. Suppose that there exists a simple oriented closed curve in Σ+ with homology class h. Let K be the push-off of this curve, framed along Σ+, with an extra ε-twist. Then, it follows from (2.5) that t =z ∈P ≃H (E(F1 );Z). K 1 Σ Remark 2.9. Accordingto the proofofLemma2.7, the Serresequencefor homology associated to the bundle F1 gives the following short exact sequence: Σ p - - - - 0 Z P H 0, 2 - whereZ isinjectedintoP bysending1to(0,1). Themaps:H P definedby 2 s(h)= h,h is a set-theoretic section. Accordingto (1.2), the associated2-cocycle - H×H Z isthe mod2reducedintersectionformofΣ. Thus,P isisomorphic 2 to H ⋊(cid:0)Z w(cid:1)ith crossed product defined by 2 (h ,ε )·(h ,ε )=(h +h ,ε +ε +h •h ). 1 1 2 2 1 2 1 2 1 2 The element h,h+ε·1 ∈P corresponds to (h,ǫ)∈H ⋊Z . 2 (cid:0) (cid:1) 2.3. The surgery map ψ 1 In this paragraph, Σ is allowed to be Σ or Σ and the surgery map ψ : g g,1 1 - A (P) C (Σ) is constructed by means of calculi of clovers. 1 1 Convention2.10. Here,weadopt Goussarov’sconventionforthesurgerymeaning of Y-graphs and clovers [4], [2].