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Abstract We compute the mapping class group orbits in the homotopy set of framings of a compact connected oriented surface with non-empty boundary. In the case g≥2 , the computation is some modification of Johnson’s results (D. Johnson, Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2)22 (1980), 365–373; D. Johnson, An abelian quotient of the mapping class group ℐg , Math. Ann.249 (1980), 225–242) and certain arguments on the Arf invariant, while we need an extra invariant for the genus 1 case. In addition, we discuss how this invariant behaves in the relative case, which Randal-Williams (O. Randal-Williams, Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces, J. Topology7 (2014), 155–186) studied for g≥2 . 1. Introduction Let Σ be a compact connected oriented smooth ( C∞ ) surface with non-empty boundary. Then the tangent bundle TΣ is a trivial bundle. Its orientation-preserving global trivializations TΣ→≅Σ×R2 are called framings of the surface Σ , which play important roles in surface topology. The mod2 reduction of a framing can be regarded as a spin structure on the surface Σ . A spin structure on a closed surface is called a theta characteristic in a classical context, and the mapping class group orbits in the set of theta characteristics are described by the Arf invariant [3]. We denote by F(Σ) the set of homotopy classes of framings of Σ , and fix a Riemannian metric ∥·∥ on the tangent bundle ϖ:TΣ→Σ . The unit tangent bundle UΣ≔{e∈TΣ;∥e∥=1}→ϖΣ is a principal S1 bundle over Σ . A framing defines a continuous map UΣ→S1 whose restriction to each fiber is homotopic to the identity 1S1 . Taking the pull-back of the positive generator of H1(S1;Z) , we obtain an element of H1(UΣ;Z) . This defines a natural embedding F(Σ)↪H1(UΣ;Z) . More precisely, F(Σ) is an affine set modeled by the abelian group ϖ*H1(Σ;Z)(≅H1(Σ;Z)) (see Section 3.1). In particular, the difference f1−f0 of two framings f0 and f1∈F(Σ) defines a unique element of H1(Σ;Z) . In this paper, we consider the mapping class group of Σ fixing the boundary pointwise M(Σ)≔π0Diff+(Σ,idon∂Σ)=Diff+(Σ,idon∂Σ)/isotopy, which acts on the set F(Σ) from the right in a natural way. If we fix an element f0∈F(Σ) , then the map k(f0):M(Σ)→H1(Σ;Z),φ↦f0◦φ−f0 is a twisted cocycle of the group M(Σ) . The cohomology class k≔[k(f0)]∈H1(M(Σ);H1(Σ;Z)) does not depend on the choice of f0 , is called the Earle class [6] or the Chillingworth class [4, 5, 15], and generates the cohomology group in the case when the boundary ∂Σ is connected and the genus of Σ is greater than 1 [11]. For the case where the boundary is not connected, see [10, Theorem 1.A]. The construction of k stated here is due to Furuta [13, Section 4]. The Morita trace [12] and its refinement, the Enomoto–Satoh trace [7], are higher analogs of the class k . In the author’s joint paper with Alekseev et al. [1], we clarify topological and Lie theoretical meanings of the Enomoto–Satoh trace. The formality problem of a variant of the Turaev cobracket for an immersed loop on the surface, the Enomoto–Satoh trace and the Kashiwara–Vergne problem in Lie theory are closely related to each other. We need the rotation number of the immersed loop with respect to a framing to define of this variant of the Turaev cobracket. This is the reason why we describe the orbit set F(Σ)/M(Σ) in this paper. The homotopy set F(Σ) we study in this paper is absolute, namely, we allow framings to move on the boundary. In fact, the rotation number of an immersed loop with respect to a framing f is invariant under any moves of f on the boundary ∂Σ . On the other hand, we can consider a relative version of the homotopy set F(Σ,δ) for a fixed framing on the boundary δ:TS∣∂Σ→≅∂Σ×R2 . Here we make framings on ∂Σ equal the given datum δ . We need the latter version to define the rotation number of an arc connecting two boundary components. Randal-Williams [14] computes the mapping class group orbits in the set of ( r-)spin structures for any genus in the relative version and those in the homotopy set F(Σ,δ) for g≥2 . It is interesting that the (generalized) Arf invariant is defined in any F(Σ,δ) [14], while it is not defined in some absolute cases as in Section 2 of this paper. In particular, the computations in this paper are different from those by Randal-Williams [14]. In the case g≥2 , the formality of the Turaev cobracket holds good for any choice of a framing. But, if g=1 , it depends on the choice of a framing, so that the formality problem is reduced to the computation of the mapping class group orbits in the set F(Σ) . It is controlled by an extra invariant A˜(f) introduced in this paper (Corollary 3.10). All these results are proved in [2]. Anyway, following Whitney [16], we consider the rotation number rotf(ℓ)∈Z of a smooth immersion ℓ:S1→Σ with respect to a framing f∈F(Σ) . We number the boundary components as ∂Σ=∐j=0n∂jΣ . The rotation numbers rotf(∂jΣ) , 0≤j≤n , are invariant under the action of the group M(Σ) . Here we endow each ∂jΣ with the orientation induced by Σ . By the Poincaré–Hopf theorem (Lemma 3.3), we have ∑j=1nrotf(∂jΣ)=χ(Σ)=1−2g−n. Our description of the orbit set F(Σ)/M(Σ) depends on the genus g(Σ) of the surface Σ . First we consider the case g(Σ)=0 . Clearly, we have Lemma 1.1 (Equation (3.7)) Suppose g(Σ)=0 . Then two framings f1and f2∈F(Σ)are homotopic to each other if and only if rotf1(∂jΣ)=rotf2(∂jΣ)for any 0≤j≤n . Next we discuss the positive genus case: g=g(Σ)≥1 . Choose a system of simple closed curves {αi,βi}i=1g on Σ as in Fig. 1. The Arf invariant of the mod2 reduction of f is defined in the case where all the numbers rotf(∂jΣ) , 0≤j≤n , are odd. Then the Arf invariant of the spin structure is defined by Arf(f)≡∑i=1g(rotf(αi)+1)(rotf(βi)+1)(mod2). Figure 1. View largeDownload slide A system of simple closed curves on Σg,n+1 . Figure 1. View largeDownload slide A system of simple closed curves on Σg,n+1 . In the case g(Σ)≥2 , we have the following. Theorem 1.2 (Theorem 3.5) Suppose g(Σ)≥2 , and f1,f2∈F(Σ) . Then f1and f2belong to the same M(Σ)-orbit, if and only if rotf1(∂jΣ)=rotf2(∂jΣ)for any 0≤j≤n . If all the numbers rotf1(∂jΣ)=rotf2(∂jΣ) , 0≤j≤n , are odd, then Arf(f1)=Arf(f2) . The proof given in Section 3.2 is some modification of Johnson’s arguments [8, 9]. The genus 1 case is different from the others. We need to introduce an invariant A˜(f)∈Z≥0 for f∈F(Σ) . It is defined to be the generator of the ideal in Z generated by the set {rotf(γ) ; γ is a non-separating simple closed curve on Σ} . We have Arf(f)≡A˜(f)+1(mod2). On the other hand, if g≥2 , we have A˜(f)=1 for any f∈F(Σ) (Lemma 3.4). Theorem 1.3 (Theorem 3.8) Suppose g(Σ)=1 , and f1,f2∈F(Σ) . Then f1and f2belong to the same M(Σ)-orbit, if and only if rotf1(∂jΣ)=rotf2(∂jΣ)for any 0≤j≤n . A˜(f1)=A˜(f2)∈Z≥0 . For the sake of non-experts on topology who are interested only in the Kashiwara–Vergne problem, this paper is self-contained except the results by Johnson [8] and Section 3.4. In particular, we will give an elementary proof of the Poincaré–Hopf theorem on the surface Σ (Lemma 3.3). In Section 2, following Johnson [8], we study the mapping class orbits in the set of spin structures on any compact surface Σ with non-empty boundary. Generalities on framings are discussed in Section 3.1. Our computation for the case g(Σ)≥2 in Section 3.2 is some modification of Johnson’s paper [9]. We need some extra invariant A˜(f) for the case g(Σ)=1 in Section 3.3. It is introduced in the end of Section 3.1. In Section 3.4, we prove that the invariant A˜(f) and the generalized Arf invariant introduced in [14] classify the mapping class group orbits in the relative genus 1 case (Theorem 3.12). In this paper, we denote by H1(−) and H1(−) the first Z-(co)homology groups, and by H1(−)(2) and H1(−)(2) the first Z/2-(co)homology groups. On H1(Σ) and H1(Σ)(2) , we have the (algebraic) intersection forms ·:H1(Σ)⊗2→Z , a⊗b↦a·b , and ·:(H1(Σ)(2))⊗2→Z/2 , a⊗b↦a·b . By the classification of surfaces, any compact connected oriented smooth surface Σ is classified by the genus and the number of the boundary components. We denote by Σg,n+1 a compact connected oriented smooth surface of genus g with n+1 boundary components for g,n≥0 . It is uniquely determined up to diffeomorphism. Throughout this paper, we fix a system of simple closed curves {αi,βi}i=1g on the surface Σg,n+1 shown in Fig. 1. By Σg,0 we mean a closed connected oriented surface of genus g . 2. Spin structures In this section, following Johnson [8], we compute the mapping class group orbits in the set of spin structures on any compact connected oriented surface Σ with non-empty boundary ∂Σ . A spin structure on Σ is, by definition, an unramified double covering of the unit tangent bundle UΣ whose restriction to each fiber is non-trivial. In a natural way, the set (of isomorphism classes) of such double coverings is isomorphic to the complement H1(UΣ)(2)⧹H1(Σ)(2) in the exact sequence 0→H1(Σ)(2)⟶ϖ*H1(UΣ)(2)⟶ι*Z/2→0 (2.1) associated with the fibration S1ι↪UΣ→ϖΣ . Here, we identify H1(Σ)(2) with its image under ϖ* . The canonical lifting H1(Σ)(2)→H1(UΣ)(2),a↦a˜, (2.2) is constructed in the same way as the original one for a closed surface by Johnson [8]. In particular, if γ:∐i=1mS1→Σ is a smooth embedding, then we have [γ]˜=γ⃗+mι*(1)∈H1(UΣ)(2), (2.3) where γ⃗:∐i=1mS1→UΣ is the (normalized) velocity vector of γ , and ι* is the dual of ι* in the sequence (2.1). As was shown in [8, Theorem 1B], we have (a+b)˜=a˜+b˜+(a·b)ι*(1) for a,b∈H1(Σ)(2) . For any ξ in the complement H1(UΣ)(2)⧹H1(Σ)(2) , a quadratic form ωξ:H1(Σ)(2)→Z/2 is defined by ωξ(a)≔⟨ξ,a˜⟩∈Z/2 for a∈H1(Σ)(2) . By a quadratic form, we mean a function H1(Σ)(2)→Z/2 satisfying ω(a+b)=ω(a)+ω(b)+a·b for any a and b∈H1(Σ)(2) . We denote by Quad(Σ) the set of quadratic forms on H1(Σ)(2) . We remark ω2−ω1:H1(Σ)(2)→Z/2 is a homomorphism, so that it can be regarded as an element of H1(Σ)(2) for any ω1 and ω2∈Quad(Σ) . More precisely, the group H1(Σ)(2) acts on the set Quad(Σ) freely and transitively, that is, the set Quad(Σ) is an affine set modeled by the abelian group H1(Σ)(2) . The mapping class group M(Σ) acts on the sets H1(UΣ)(2)⧹H1(Σ)(2) and Quad(Σ) in a natural way. The map ξ↦ωξ defines an M(Σ)-equivariant isomorphism between the sets H1(UΣ)(2)⧹H1(Σ)(2) and Quad(Σ) . For the rest of this section, we compute the mapping class group orbits in the set of quadratic forms, Quad(Σ) . We begin by recalling some elementary facts on the (co)homology of the surface Σ . The cohomology exact sequence H1(Σ,∂Σ)(2)⟶j*H1(Σ)(2)⟶i*H1(∂Σ)(2) (2.4) is compatible with the action of the mapping class group M(Σ) . In particular, the subgroup imj*=keri*⊂H1(Σ)(2) is stable under the action of M(Σ) , and equals the image of the map H1(Σ)(2)→H1(Σ)(2) , x↦x· , from the Poincaré–Lefschetz duality. Lemma 2.1 Any homology class in H1(Σ)(2)is represented by a simple closed curve. Proof The four elements in H1(Σ1,0)(2) are represented by simple closed curves. Similarly, all elements in H1(Σ0,n+1)(2) are represented by simple closed curves. Any element in H1(Σg,n+1)(2) can be represented by the connected sum of some of these elements. This proves the lemma.□ For any a∈H1(Σ)(2) we introduce a map Ta:H1(Σ)(2)→H1(Σ)(2) defined by x↦x−(x·a)a . If γ represents the element a , the map Ta is induced by the right-handed Dehn twist along γ denoted by tγ∈M(Σ) . In particular, Ta respects the intersection form. We denote by G(Σ)⊂Aut(H1(Σ)(2)) the subgroup generated by {Ta;a∈H1(Σ)(2)} . From the Dehn–Lickorish theorem and Lemma 2.1, it equals the image of the mapping class group M(Σ) in the group Aut(H1(Σ)(2)) . In particular, the M(Σ)-orbits in the set Quad(Σ) are the same as the G(Σ)-orbits. For a quadratic form ω:H1(Σ)(2)→Z/2 , we define a map mω:G(Σ)→H1(Σ)(2) by S↦mω(S)≔ωS−ω . Then, we have mω(S1S2)=mω(S1)S2+mω(S2) (2.5) for any S1 and S2∈G(Σ) . One can compute ⟨mω(Ta),x⟩=ω(Tax)−ω(x)=ω(x−(x·a)a)−ω(x)=(x·a)ω(a)+(x·a)2=(x·a)(ω(a)+1) for a,x∈H1(Σ)(2) . This means mω(Ta)=(ω(a)+1)a·∈imj*⊂H1(Σ)(2). (2.6) Hence, we obtain a 1-cocycle mω:G(Σ)→imj*(⊂H1(Σ)(2)) . Theorem 2.2 Let ω1and ω2:H1(Σ)(2)→Z/2be quadratic forms. Then, ω1and ω2belong to the same M(Σ)-orbit if and only if ∃x∈H1(Σ)(2)s.t.ω1(x)=0,ω2−ω1=x·∈imj*. (♯) Proof We denote by ω1∼ω2 the assertion that ω1 and ω2 satisfy the condition ( ♯ ), and begin the proof by checking that the relation ∼ is an equivalence relation on the set Quad(Σ) . The reflexivity ω∼ω follows from ω(0)=0 . If x∈H1(Σ)(2) satisfies ω1(x)=0 , then we have (ω1+x·)(x)=ω1(x)+x·x=0 , which proves the symmetry: (ω1∼ω2)⇒(ω2∼ω1) . Assume ω1∼ω2 and ω2∼ω3 . This means there exist x1 and x2∈H1(Σ)(2) such that ω1(x1)=ω2(x2)=0 , ω2−ω1=x1· and ω3−ω2=x2· . Then we have ω3−ω1=(x1+x2)· and ω1(x1+x2)=ω1(x1)+x1·x2+ω1(x2)=ω1(x1)+ω2(x2)=0 . Hence, we obtain ω1∼ω3 . This proves the transitivity. Next we assume ω2=ω1Ta for some a∈H1(Σ)(2) . Then, by formula (2.6), we have ω2−ω1=mω1(Ta)=(ω1(a)+1)a· , while ω1((ω1(a)+1)a)=(ω1(a)+1)ω1(a)=0 . This implies ω1∼ω1Ta . The relation ∼ is an equivalence relation, and G(Σ) is generated by Ta’s. Hence, if ω1 and ω2 belong to the same G(Σ)-orbit, then we have ω1∼ω2 . Conversely, if there exists some x∈H1(Σ)(2) such that ω1(x)=0 and ω2−ω1=x· . Then we have ω1Tx−ω1=mω1(Tx)=(ω1(x)+1)x·=x·=ω2−ω1 , so that ω2=ω1Tx . In particular, ω1 and ω2 belong to the same G(Σ)-orbit. This completes the proof of the theorem.□ Now consider the inclusion homomorphism i*:H1(∂Σ)(2)→H1(Σ)(2) . Any ω∈Quad(Σ) restricts to a homomorphism on H1(Σ)(2) via the homomorphism i* , since the intersection form vanishes on i*H1(∂Σ)(2) . Hence, we have the restriction map i*:Quad(Σ)→H1(∂Σ)(2),ω↦i*ω=ω◦i*. (2.7) The kernel keri* is spanned by the Z/2-fundamental class [∂Σ]2∈H1(∂Σ)(2) . Hence, if h∈H1(∂Σ)(2) satisfies h[∂Σ]2=0 , then it induces a homomorphism on i*H1(∂Σ)(2) , and extended to the element of H1(Σ)(2) satisfying h([αi])=h([βi])=0 for any 1≤i≤g . Here αi and βi are the simple closed curves shown in Fig. 1. Moreover, we define a map ω0,h:H1(Σ)(2)→Z/2 by ω0,h(x)≔∑i=1g(x·[αi])(x·[βi])+h(x) (2.8) for x∈H1(Σ)(2) . It is easy to check that ω0,h is a quadratic form, and i*ω0,h=h . If a quadratic form ω∈Quad(Σ) satisfies i*ω=0∈H1(∂Σ)(2) , then the Arf invariant Arf(ω) is defined by Arf(ω)≔∑i=1gω([αi])ω([βi])∈Z/2 (2.9) [3] . For any x∈H1(Σ)(2) , we have Arf(ω0,0+x·)=∑i=1g(x·[αi])(x·[βi])=ω0,0(x). (2.10) In particular, the Arf invariant Arf is G(Σ)-invariant, namely, we have Arf(ωS)=Arf(ω) for any ω∈(i*)−1(0) and S∈G(Σ) . In fact, there are x0 and x1∈H1(Σ)(2) such that ω=ω0,0+x0· , ωS−ω=x1· and ω(x1)=0 . Then we have Arf(ωS)=ω0,0(x0+x1)=ω0,0(x0)+x0·x1+ω0,0(x1)=Arf(ω)+ω(x1)=Arf(ω) . Now recall mω(G(Σ))⊂ker(i*:H1(Σ)(2)→H1(∂Σ)(2)) and G(Σ) is the image of M(Σ) in Aut(H1(Σ)(2)) . Hence, the restriction map i* induces the map ρ2:Quad(Σ)/M(Σ)→H1(∂Σ)(2),ωmodG(Σ)↦i*ω. (2.11) Theorem 2.3 For any h∈H1(∂Σ)(2) , the cardinality of the set ρ2−1(h)is given by ♯ρ2−1(h)={0ifh[∂Σ]2≠0,1ifh[∂Σ]2=0and(h≠0org=0),2ifh=0andg≥1.In the last case, the two orbits are distinguished by the Arf invariant Arf:(i*)−1(0)→Z/2 . Proof (0) If h[∂Σ]2≠0 , we have (i*)−1(h)=∅ since i*[∂Σ]2=0 . (1) Suppose h[∂Σ]2=0 and g=0 . Then, (i*)−1(h)={ω0,h} is a one-point set. Next suppose h[∂Σ]2=0 , h≠0 and g≥1 . Then ω0,h∈(i*)−1(h)≠∅ . For any ω∈(i*)−1(h) , we have ω−ω0,h∈keri*=imj* , so that ω−ω0,h=x0·∈H1(Σ)(2) for some x0∈H1(Σ)(2) . Since h≠0 , we have ω(x0)=h(x1) for some x1∈H1(∂Σ)(2) . Then (x0+x1)·=x0·=ω−ω0,h and ω(x0+x1)=ω(x0)+x0·x1+ω(x1)=h(x1)+0+h(x1)=0 . By Theorem 2.2, we have ω=ω0,hS for some S∈G(Σ) . This proves ♯ρ2−1(h)=1 . (2) Suppose h=0 and g≥1 . Then ω0,0∈(i*)−1(0)≠∅ , and we have ω0,0(x0)=1 for some x0∈H1(Σ)(2) . For any ω∈(i*)−1(0) , there exists some x∈H1(Σ)(2) such that ω−ω0,0=x·∈H1(Σ)(2) . If ω0,0(x)=Arf(ω)=0 , then, by Theorem 2.2, we have ω=ω0,0S for some S∈G(Σ) . On the other hand, if ω0,0(x)=Arf(ω)=1 , then we have ω−(ω0,0+x0·)=(x−x0)· and (ω0,0+x0·)(x−x0)=ω0,0(x−x0)+x0·x=ω0,0(x)−ω0,0(x0)=0 . This implies ω=(ω0,0+x0·)S for some S∈G(Σ) . This proves ♯ρ2−1(0)=2 . This completes the proof of the theorem.□ As was proved by Randal-Williams in [14, Theorem 2.9], the cardinality of the mapping class group orbit sets in the set of spin structures for the relative version is always 2, and does not depend on the boundary value. In particular, the (generalized) Arf invariant can be defined in any cases. The situation is similar for framings in the case g≥2 (Theorem 3.5). 3. Framings 3.1. Generalities Let Σ be a compact connected oriented smooth surface with non-empty boundary as before. In this paper, we denote by F(Σ) the set of homotopy classes of framings, that is, orientation-preserving global trivializations TΣ→≅Σ×R2 of the tangent bundle TΣ . In this paper, the composite of such an trivialization and the second projection, TΣ→≅Σ×R2→pr2R2 , is also called a framing. The group [Σ,S1]=H1(Σ)=H1(Σ;Z) acts on the set F(Σ) freely and transitively. In fact, the difference of any two framings gives a continuous map Σ→GL+(2;R)≃S1 . The mapping class group M(Σ) acts on the set F(Σ) from the right in a natural way. Consider the inclusion map ι:S1↪UΣ and the projection ϖ:UΣ→Σ as in the preceding section. Then, we have M(Σ)-equivariant exact sequences 0→Z⟶ι*H1(UΣ)⟶ϖ*H1(Σ)→0,and (3.1) 0→H1(Σ)⟶ϖ*H1(UΣ)⟶ι*Z→0 (3.2) in the integral (co)homology. The group H1(Σ) obviously acts on the inverse image (ι*)−1(1) of 1∈Z freely and transitively. For a framing f∈F(Σ) , we denote by ξ(f)∈H1(UΣ) the pull-back of the positive generator of H1(S1) by the map f:UΣ→S1 . It is clear that ι*ξ(f)=1∈Z . Then the map F(Σ)→(ι*)−1(1) , f↦ξ(f) , is equivariant under the actions of the groups M(Σ) and H1(Σ) . In particular, it is an M(Σ)-equivariant isomorphism F(Σ)≅(ι*)−1(1) , by which we identify these two sets with each other. An immersion ℓ:S1→Σ lifts to its (normalized) velocity vector ℓ⃗:S1→UΣ , t↦ℓ˙(t)/∥ℓ˙(t)∥ . The rotation number of ℓ with respect to a framing f is defined by rotfℓ≔⟨ξ(f),[ℓ⃗]⟩=deg(f◦ℓ⃗:S1→S1)∈Z (3.3) [16] . For any φ∈M(Σ) , we have rotf◦φ(ℓ)=rotf(φ◦ℓ). (3.4) Lemma 3.1 If ℓi:S1→Σ , 1≤i≤b1=b1(Σ) , is an immersion, and the set {[ℓi]}i=1b1constitutes a free basis of H1(Σ) , then the map F(Σ)→Zb1,f↦(rotf(ℓi))i=1b1is a bijection. Proof Then the set {[ℓi⃗]}i=1b1∪{ι*(1)} constitutes a free basis of H1(UΣ) .□ The mod 2 reduction of ξ(f) , which we denote by ξ2(f)∈H1(UΣ)(2) , is a spin structure on the surface Σ . We write simply ωf≔ωξ2(f):H1(Σ)(2)→Z/2 for the corresponding quadratic form. Lemma 3.2 For any smooth embedding ℓ:S1→Σ , we have ωf([ℓ])=rotf(ℓ)+1∈Z/2. Proof Recall the canonical lifting in [8] is given by [ℓ]˜=[ℓ⃗]+ι*(1)∈H1(UΣ)(2) . Hence, we have ωf([ℓ])=⟨ξ2(f),[ℓ]˜⟩=⟨ξ2(f),[ℓ⃗]⟩+1=rotf(ℓ)+1∈Z/2. This proves the lemma.□ The following is a straight-forward consequence of the Poincaré–Hopf theorem. But we will give its elementary proof for the convenience of non-experts on topology. Lemma 3.3 Let S⊂Σbe a compact smooth subsurface. We number the boundary components of S : ∂S=∐k=1N∂kS . Then, we have ∑k=1Nrotf(∂kS)=χ(S)for any f∈F(Σ) . Here we endow each ∂kSwith the orientation induced by S , and χ(S)is the Euler characteristic of the surface S. Proof Let {(eλ,φλ:Dnλ→S)}λ∈Λ be a finite cell decomposition of the surface S such that each characteristic map φλ:Dnλ→eλ¯⊂S is a smooth embedding and each 0-cell is located on the boundary ∂S . We denote Ci≔♯{λ∈Λ;nλ=i} , 0≤i≤2 , so that χ(S)=C2−C1+C0 . Then we compute the sum ∑nλ=2rotf(φλ(∂Dnλ)) . Since the loop φλ(∂D2) is regular homotopic to a small loop around the center of eλ , the sum equals C2 . The contribution of both sides of each interior 1-cell cancels each other, while the contribution of the boundary 1-cells equals the sum ∑k=1Nrotf(∂kS) . The contribution of a vertex, that is, a 0-cell eλ equals 12(dλ−2) , where dλ is the valency at the vertex eλ . See Fig. 2. On the other hand, we have C1=12∑nλ=0dλ . Hence, we obtain C2=(∑k=1Nrotf(∂kS))+12∑nλ=0(dλ−2)=(∑k=1Nrotf(∂kS))+C1−C0, which proves the lemma.□ Figure 2. View largeDownload slide The case dλ=5 . Figure 2. View largeDownload slide The case dλ=5 . Now suppose Σ=Σg,n+1 for g,n≥0 . We number the boundary components: ∂Σ=∐i=0n∂iΣ . Since any element of the group M(Σ) fixes the boundary pointwise, we can define a map ρ:F(Σ)/M(Σ)→Zn+1,fmodM(Σ)↦(rotf(∂iΣ)+1)i=0n. (3.5) Here, taking Lemma 3.2 into account, we consider rotf(∂iΣ)+1 instead of the rotation number itself. By Lemmas 3.1 and 3.3, we have imρ={(νi)i=0n∈Zn+1;∑i=0nνi=2−2g}. (3.6) In the genus 0 case, that is, Σ=Σ0,n+1 , these lemmas imply F(Σ)/M(Σ)=F(Σ)≅ρ{(νi)i=0n∈Zn+1;∑i=0nνi=2}. (3.7) We conclude this subsection by introducing an extra invariant for a framing, which will be used for the genus 1 case. For f∈F(Σ) , we consider the ideal a(f) in Z generated by the set {rotf(γ) ; γ is a non-separating simple closed curve on Σ} , and define A˜(f)∈Z≥0 to be the non-negative generator of the ideal a(f) . It is clear that these are invariants under the action of the mapping class group M(Σ) . But, if g≥2 , they are trivial invariants. Lemma 3.4 If g≥2 , we have A˜(f)=1for any f∈F(Σg,n+1) . Proof From the assumption, there is a smooth compact subsurface P⊂Σ diffeomorphic to a pair of pants Σ0,3 such that each of the three boundary components ∂iP , 0≤i≤2 , is a non-separating curve in Σ . Then, from Lemma 3.3, we have rotf(∂0P)+rotf(∂1P)+rotf(∂2P)=χ(P)=−1 , so that −1∈a(f) . This proves the lemma.□ 3.2. The case g≥2 In this subsection, we consider Σ=Σg,n+1 for the case g≥2 . In this case, our computation modifies that in [9]. Consider the map ρ:F(Σ)/M(Σ)→Zn+1 in (3.5). Theorem 3.5 Suppose g≥2 . Then, for any ν∈imρ={(νi)i=0n∈Zn+1;∑i=0nνi=2−2g} , wehave ♯ρ−1(ν)={1ifν∈imρ⧹(2Z)n+1,2ifν∈imρ∩(2Z)n+1.In the latter case, the two orbits are distinguished by the Arf invariant of the spin structure ξ2(f) . Proof Let f1 and f2∈F(Σ) satisfy ρ(f1)=ρ(f2) and Arf(ξ2(f1))=Arf(ξ2(f2)) if ρ(f1)=ρ(f2)∈(2Z)n+1 . Then, by Theorem 2.3, we have ξ(f2)−ξ(f1φ0)=2(∑i=1gλi[αi]+∑i=1gμi[βi])·∈H1(Σ) (3.8) for some φ0∈M(Σ) and λi,μi∈Z . Here αi and βi are the simple closed curves shown in Fig. 1. Hence, it suffices to construct φi′ and φi″∈M(Σ) for each 1≤i≤g such that ξ(fφi′)−ξ(f)=2[αi]·andξ(fφi″)−ξ(f)=2[βi] (3.9) for any f∈F(Σ) . We denote by tγ∈M(Σ) the right-handed Dehn twist along a simple closed curve γ in Σ . Now from the assumption g≥2 , there exist simple closed curves αˆi and βˆi satisfying the conditions αi and αˆi bound a smooth compact subsurface diffeomorphic to Σ1,2 . βi and βˆi bound a smooth compact subsurface diffeomorphic to Σ1,2 . αˆi and βˆi are disjoint from {αk,βk}k≠i . αˆi intersects with βi transversely at a unique point. βˆi intersects with αi transversely at a unique point. Choose a point on each component of ∂Σ1,2 . Then, by the disk theorem, two simple arcs connecting these two chosen points are mapped to each other by the action of the group M(Σ1,2) . Similar transitivity holds also for the surface Σg−2,n+3 . Hence, by the classification theorem of surfaces, the quadruples (Σ,αi,αˆi,βi) and (Σ,βi,βˆi,αi) are diffeomorphic to (Σ,γ1,γ2,γ0) in Fig. 3(a). Then the simple closed curve tγ2−1tγ1(γ0) is computed as in Fig. 3(b), so that γ0 and tγ2−1tγ1(γ0) bound a smooth compact subsurface diffeomorphic to Σ1,2 By Lemma 3.3, we have ∣rotftγ2−1tγ1(γ0)−rotf(γ0)∣=∣χ(Σ1,2)∣=2 for any f∈F(Σ) . It is clear that rotftγ2−1tγ1(γ1)=rotf(γ1) . The mapping class tγ2−1tγ1 is just a BP-map in [9]. Hence, if we take φi′ to be tαˆi−1tαi or its inverse, then rotfφi′(βi)−rotf(βi)=2 and rotfφi′(αi)−rotf(αi)=0 . From the condition (ii) above, rotfφi′(αk)−rotf(αk)=0 and rotfφi′(βk)−rotf(βk)=0 for k≠i . Hence, ξ(fφi′)−ξ(f)=2[αi]· as desired in (3.9). Similarly, if we take φi″ to be tβi−1tβi or its inverse, then φi″ satisfies (3.9). This proves the theorem.□ Figure 3. View largeDownload slide Computation of tγ2−1tγ1(γ0) . Figure 3. View largeDownload slide Computation of tγ2−1tγ1(γ0) . 3.3. The genus 1 case Finally, we study the genus 1 case: Σ=Σ1,n+1 . We write simply α=α1 and β=β1 shown in Fig. 1, νj=νj(f)≔rotf(∂jΣ)+1 , 0≤j≤n , and take a closed regular neighborhood Σ′ of the subset α(S1)∪β(S1) . It is diffeomorphic to Σ1,1 . We begin by computing the invariant A˜(f) for f∈F(Σ) . Lemma 3.6 The ideal in Zgenerated by the set {rotf(α),rotf(β),νj(f);0≤j≤n}equals the ideal a(f) . In other words, A˜(f)is the non-negative greatest common divisor of the set. Proof We denote the ideal given above by b(f) . For each 0≤j≤n , we choose a band connecting α and ∂jΣ to obtain a non-separating simple closed curve α(j) such that α , ∂jΣ and α(j) bound a pair of pants. Then we have rotf(α(j))=rotf(α)+νj , so that we obtain b(f)⊂a(f) . Let γ be any non-separating simple closed curve in Σ . When the curve γ crosses the boundary component ∂jΣ , the rotation number changes by ±(rotf(∂jΣ)+1)=±νj . Hence, there exists a non-separating simple closed curve γ′ in Σ′ such that rotf(γ)−rotf(γ′)∈b(f) . The curve γ′ is mapped to α by an element of the subgroup generated by the Dehn twists tα and tβ . For any simple closed curve γ″ in Σ , we have rotf(tβ(γ″))−rotf(γ″)=([γ″]·[β])rotf(β)∈b(f) (3.10) and rotf(tα(γ″))−rotf(γ″)∈b(f) . Hence, we have rotf(γ′)∈rotf(α)+b(f)=b(f) . This proves a(f)⊂b(f) , and completes the proof of the lemma.□ Corollary 3.7 If rotf(∂jΣ)is odd for any 0≤j≤n , we have Arf(ξ2(f))≡A˜(f)+1(mod2). Proof By Lemma 3.2, we have Arf(ξ2(f))≡(rotf(α)+1)(rotf(β)+1)(mod2) .□ Theorem 3.8 Suppose g=1 , and f1,f2∈F(Σ1,n+1) . Then f1and f2belong to the same M(Σ1,n+1)-orbit if and only if f1and f2satisfy both of the following conditions: rotf1(∂jΣ)=rotf2(∂jΣ)for any 0≤j≤n , and A˜(f1)=A˜(f2)∈Z≥0 . Proof If f1 and f2 belong to the same M(Σ)-orbit, then it is clear that they satisfy both of the conditions. Hence, it suffices to prove the following: for any f∈F(Σ) , we have (rotfφ(α),rotfφ(β))=(A˜(f),0)∈Z2 for some φ∈M(Σ) . From formula (3.10) and the similar one for tα , the actions of tα and tβ on the row vectors (rotf(α),rotf(β))∈Z2 generate the standard right action of SL2(Z) on Z2 . By the Euclidean algorithm, the vectors (a1,b1) and (a2,b2)∈Z2 belong to the same SL2(Z)-orbit if and only if gcd(a1,b1)=gcd(a2,b2)∈Z . We denote d≔gcd(rotf(α),rotf(β)) and c≔gcd(νj(f);0≤j≤n) . Then A˜(f)=gcd(c,d) . By the Euclidean algorithm, we have (rotfφ1(α),rotfφ1(β))=(d,0) for some φ1∈M(Σ) . Recall the non-separating simple closed curve α(j) introduced in the proof of Lemma 3.6. For any f′∈F(Σ) , we have rotf′(tα−1tα(j)(α))=rotf′(α),androtf′(tα−1tα(j)(β))=rotf′(tα(j)(β))+([β]·[α])rotf′(α)=rotf′(β)−([β]·[α])rotf′(α(j))+([β]·[α])rotf′(α)=rotf′(β)−νj(f′). Hence, there exists an element φ2 in the subgroup generated by the elements tα−1tα(j) , 0≤j≤n , such that (rotfφ1φ2(α),rotfφ1φ2(β))=(d,c) . Recall A˜(f)=gcd(c,d) . By the Euclidean algorithm, we have (rotfφ1φ2φ3(α),rotfφ1φ2φ3(β))=(A˜(f),0) for some φ3∈M(Σ) . This proves the theorem.□ Corollary 3.9 For ν=(νj)j=0n∈Zn+1⧹{0}with ∑j=0nνj=0 , the inverse image ρ−1(ν)is parametrized by the positive divisors of gcd(νj;0≤j≤n) , while ρ−1(0)by the non-negative integers Z≥0 . Proof If ν≠0 , then A˜(f) is a positive divisor of the gcd . The corollary follows from Lemma 3.1. See also Equation (3.6).□ The following is related to the formality problem of the Turaev cobracket on genus 1 surfaces [1]. Corollary 3.10 For f∈F(Σ1,n+1) , there exists a mapping class φ∈M(Σ1,n+1)satisfying rotfφ(α)=rotfφ(β)=0 , if and only if A˜(f)=gcd(νj;0≤j≤n) . Proof By Lemma 3.1, there exists a framing f•∈F(Σ1,n+1) such that rotf•(α)=rotf•(β)=0 and νj(f•)=νj(f) for any 0≤j≤n . Then A˜(f•)=gcd(νj;0≤j≤n) from Lemma 3.6. Hence, the corollary follows from Theorem 3.8.□ The following theorem is proved in [2]. Theorem 3.11 ([2]). The Kashiwara–Vergne (KV) problem for the surface Σg,n+1with respect to a framing fhas a solution if and only if g≥2 , g=1and νj≠0for some 0≤j≤n , or g=1and rotf(α)=rotf(β)=νj=0for any 0≤j≤n . This matches the results in the present paper. In fact, the KV problem is defined over a field of characteristic 0. The mapping class orbits in the framings are finite in the cases (i) and (ii), while those are infinite if g=1 and νj=0 for any 0≤j≤n . In the last case, the necessary and sufficient condition for the KV problem with respect to f to have a solution is exactly A˜(f)=0∈Z . 3.4. The relative genus 1 case We conclude this paper by some discussion about the relative version [14], which we will need to describe to the self-intersection of an immersed path. Here we fix a framing of the tangent bundle restricted to the boundary δ:TΣ∣∂Σ→≅∂Σ×R2 , and consider the set F(Σ,δ) of homotopy classes of framings f:TΣ→≅Σ×R2 which extend the framing δ , where all the homotopies we consider fix δ pointwise. By Lemma 3.3 and some obstruction theory, the set F(Σ,δ) is not empty if and only if ∑j=0nrotδ(∂jΣ)=χ(Σ) . For the rest, we assume F(Σ,δ)≠∅ . In this setting, for any f∈F(Σ,δ) , we can consider the rotation number rotf(ℓ)∈R of an immersed path ℓ connecting two different points on the boundary ∂Σ . We denote by 1∈S1 the unit element of S1=SO(2) . The group [(Σ,∂Σ),(S1,1)]=H1(Σ,∂Σ;Z)=H1(Σ,∂Σ) acts on the set F(Σ,δ) freely and transitively. For 1≤j≤n , we choose a point ∗j∈∂jΣ and a simple arc ηj from a point on ∂0Σ to ∗j such that each ηj is disjoint from {αi,βi}i=1g∪{ηk}k≠j , and transverse to ∂0Σ and ∂jΣ . Then the homology classes {[αi],[βi]}i=1g∪{[ηj]}j=1n constitute a free basis of H1(Σ,∂Σ) . The evaluation map Ev:F(Σ,δ)→Z2g+n,f↦((rotf(αi),rotf(βi))i=1g,(⌈rotf(ηj)⌉)j=1n) (3.11) is bijective, and compatible with the action of H1(Σ,∂Σ) . Here ⌈rotf(ηj)⌉∈Z is the ceiling of the rotation number rotf(ηj)∈R . Randal-Williams [14] introduced the generalized Arf invariant Arf^(f)∈Z/2 by Arf^(f)≔∑i=1g(rotf(αi)+1)(rotf(βi)+1)+∑j=1nνj⌈rotf(ηj)⌉mod2∈Z/2, (3.12) which is denoted by A(f) in the original paper [14]. The mapping class group M(Σ) acts on the set F(Σ,δ) in a natural way. As was proved in [14], the generalized Arf invariant is invariant under the mapping class group action for any g≥0 , and, if g≥2 , the orbit set F(Σ,δ)/M(Σ) is of cardinality 2 or 0 for any δ , and described by the generalized Arf invariant. Now we consider the case g=1 . We use the notation in Section 3.3. The invariant A˜(f) is related to the generalized Arf invariant Arf^(f) as follows: Suppose A˜(f) is even. Then rotf(α) , rotf(β) and all of νj’s are even. Hence, Arf^(f)≡(rotf(α)+1)(rotf(β)+1)≡1mod2 . If f1∈F(Σ,δ) is given by Ev(f1)=((A˜(f),0),(0,…,0)) , then we have A˜(f1)=A˜(f) . Next we consider the case A˜(f) is odd and Arf^(f)=0mod2 . If f2∈F(Σ,δ) is given by Ev(f2)=((A˜(f),0),(0,…,0)) , then we have A˜(f2)=A˜(f) and Arf^(f2)=0mod2 . Finally assume that A˜(f) is even and Arf^(f)=1mod2 . Then we have νj≡1(mod2) for some 1≤j≤n . If not, rotf(α) or rotf(β) are odd, so that Arf^(f)=0mod2 . This contradicts the assumption. Let j0 be the maximum j satisfying νj≡1(mod2) . If f3∈F(Σ,δ) is given by Ev(f3)=((A˜(f),0),(0,…,0,1˘j0,0,…,0)) , then we have A˜(f3)=A˜(f) and Arf^(f3)=1mod2 . From Lemma 3.6, the invariant A˜(f) can be realized to be any non-negative divisor of gcd(νj;0≤j≤n) . Here we agree that any integer is a divisor of 0. Theorem 3.12 Suppose g=1and F(Σ,δ)≠∅ . Then the orbit set F(Σ,δ)/M(Σ)is parametrized by the invariant A˜(f)and the generalized Arf invariant Arf^(f) . More precisely, for any f∈F(Σ) , we have f=fk◦φfor some φ∈M(Σ)and k=1,2,3 . Here we choose fkaccording to the invariants A˜(f)and Arf^(f)as stated above. Proof We may assume that each ηj is disjoint from the subsurface Σ′(≅Σ1,1) , a regular neighborhood of α(S1)∪β(S1) . There is an element τ∈M(Σ) whose support is in Σ′ such that (rotf◦τ(α),rotf◦τ(β))=(−rotf(α),−rotf(β)) for any f∈F(Σ,δ) . In fact, τ can be obtained as some product of Dehn twists tα and tβ . In particular, we have rotf◦τ(ηj)=rotf(ηj) for any 0≤j≤n . Next we consider a framing f∈F(Σ,δ) which satisfies Ev(f)=((A,0),(ρ1,…,ρn)) for some ρj∈Z . Here we assume A=A˜(f) . We remark that A divides any νj , 0≤j≤n . Recall the non-separating simple closed curve α(j) introduced in the proof of Lemma 3.6. Here we choose the band connecting α and ∂jΣ to be disjoint from any ηk , 1≤k≤n . Then, the curve α(j) is disjoint from ηk for k≠j , and we may assume that α(j) and ηj intersect transversely to each other at the unique point. We define ψj≔tα(j)tα−1−(νj/A)t∂jΣ−1∈M(Σ) . Since rotf(α(j))=A+νj and rotf(∂jΣ)=νj−1 , we have rotf◦ψj(ηj)−rotf(ηj)=−A−νj+νj−1=−A−1 and rotf◦ψj(β)−rotf(β)=A+νj−(1+(νj/A))A=0 . Clearly we have rotf◦ψj(α)=rotf(α)=A and rotf◦ψj(ηk)−rotf(ηk)=0 for k≠j . Moreover, we define ψj′≔τtα(j)tα−1+(νj/A)t∂jΣ−1τ−1∈M(Σ) . Similarly, we have rotf◦ψj′(ηj)−rotf(ηj)=A−1 , rotf◦ψj′(α)=rotf(α)=A , rotf◦ψj′(β)=rotf(β) and rotf◦ψj′(ηk)−rotf(ηk)=0 for k≠j . As a consequence of the construction of ψj and ψj′ , there is some φj∈M(Σ) and ϵj∈{0,1} such that Ev(f◦φj)=((A,0),(ρ1,…,ρj−1,ϵj,ρj+1,…,ρn)) and ϵj≡ρj(mod2) . In fact, gcd{−A−1,A−1} divides 2. Now we consider an arbitrary element f0∈F(Σ,δ) . We denote A=A˜(f0) . From the proof of Theorem 3.8, we have Ev(f0◦φ0)=((A,0),(ρ10,…,ρn0)) for some φ0∈M(Σ) and ρj0∈Z : Suppose A=A˜(f) is even. Then we may assume each ρj0 is even. In fact, rotf◦t∂jΣ(ηj)−rotf(ηj)=−rotf(∂jΣ)=−νj+1 is odd for any f∈F(Σ,δ) . Hence, we have some suitable product φ˜∈M(Σ) of φj∈M(Σ)’s stated above such that Ev(f0◦φ0◦φ˜)=((A,0),(0,…,0)) . This means f0◦φ0◦φ˜=f1∈F(Σ,δ) , as was desired. Assume that A=A˜(f) is odd and Arf^(f)=0mod2 . Then we have 0=Arf^(f0)=Arf^(f0◦φ0)≡A+1+∑j=1nνj⌈rotf0◦φ0(ηj)⌉≡∑j=1nνj⌈rotf0◦φ0(ηj)⌉(mod2) . Hence, there are some 1≤j1<j2<⋯<j2m≤n such that νjs⌈rotf0◦φ0(ηjs)⌉≡1(mod2) and νj⌈rotf0◦φ0(ηj)⌉≡0(mod2) if j∉{j1,j2,…,j2m} . We choose a band connecting ∂j1(Σ) and ∂j2(Σ) disjoint from α , β and ηk for k≠j1,j2 to obtain a separating simple closed curve λ such that ∂j1(Σ) , ∂j2(Σ) and λ bound a pair of pants. Then rotf0◦φ0(λ)=νj1+νj2−1 is odd. Hence, we have ⌈rotf0◦φ0◦tλ(ηj1)⌉≡⌈rotf0◦φ0◦tλ(ηj2)⌉≡0(mod2) . By similar consideration, we obtain some φ′∈M(Σ) such that ⌈rotf0◦φ0◦φ′(ηj)⌉≡0(mod2) for any 1≤j≤n . Hence, we have some suitable product φ˜∈M(Σ) of φj∈M(Σ)’s such that Ev(f0◦φ0◦φ′◦φ˜)=((A,0),(0,…,0)) . This means f0◦φ0◦φ′◦φ˜=f2∈F(Σ,δ) , as was desired. Assume A=A˜(f) is odd and Arf^(f)=1mod2 . Then ∑j=1nνj⌈rotf0◦φ0(ηj)⌉≡1(mod2) . Hence, there are some 1≤j1<j2<⋯<j2m−1≤n such that νjs⌈rotf0◦φ0(ηjs)⌉≡1(mod2) and νj⌈rotf0◦φ0(ηj)⌉≡0(mod2) if j∉{j1,j2,…,j2m−1} . In a similar way to (2), we obtain some φ′∈M(Σ) such that ⌈rotf0◦φ0◦φ′(ηj0)⌉≡1(mod2) and ⌈rotf0◦φ0◦φ′(ηj)⌉≡0(mod2) for any j≠j0 . Hence, we have some suitable product φ˜∈M(Σ) of φj∈M(Σ)ʼs such that Ev(f0◦φ0◦φ′◦φ˜)=((A,0),(0,…,0,1˘j0,0,…,0)) . This means f0◦φ0◦φ′◦φ˜=f3∈F(Σ,δ) , as was desired. This completes the proof of the theorem.□ The situation for the relative genus 0 case is elementary, but seems too complicated to describe by some simple invariants. Funding The present research is partially supported by the Grant-in-Aid for Scientific Research (S) (No. 24224002) and (B) (No. 15H03617) from the Japan Society for Promotion of Sciences. Acknowledgements This paper is a byproduct of the author's joint work with Anton Alekseev, Yusuke Kuno and Florian Naef. In particular, it has its own origin in Alekseev's question to the author. First of all the author thanks all of them for helpful discussions. Furthermore, Kuno kindly prepared all the figures in this paper. After the first draft of this paper was uploaded at the arXiv, Oscar Randal-Williams let the author know his results in [14]. Let me remark that I thank Alekseev, Kuno and Naef as well as Randal-Williams and Putman in this paper. References 1 A. Alekseev , N. Kawazumi , Y. Kuno and F. Naef , Higher genus Kashiwara–Vergne problems and the Goldman–Turaev Lie bialgebra , C. R. Acad. Sci. Paris Ser. I. 355 ( 2017 ), 123 – 127 . Google Scholar Crossref Search ADS 2 A. Alekseev , N. Kawazumi , Y. Kuno and F. Naef , The Goldman–Turaev Lie bialgebra and the Kashiwara–Vergne problem in higher genera, in preparation. 3 C. Arf , Untersuchungen über quadratische Formen in Körpern der Charakteristik 2 , J. Reine Angew. Math. 183 ( 1941 ), 148 – 167 . 4 D. R. J. 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The Quarterly Journal of Mathematics – Oxford University Press
Published: Dec 1, 2018
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