EMBEDDINGS OF BRAID GROUPS INTO MAPPING CLASS GROUPS AND THEIR HOMOLOGY Carl-Friedrich B¨odigheimer and Ulrike Tillmann Abstract We construct several families of embeddings of braid groups into mapping class groups of orientable and non-orientable surfaces and prove that they induce the trivial map in stable homology in the orientable case, but not so in the non-orientable case. We show that these embeddings are non- geometric in the sense that the standard generators of the braid group are not mapped to Dehn twists. 1 Introduction Let Γg,n denote the mapping class group of an oriented surface Σg,n of genus g with n parametrized boundary components, i.e., Γg,n is the group of connected components of the group of orientation preserving diffeomorphisms of Σg,n that fix the boundary point-wise. For a simple closed curve a on the surface, let Da denote the Dehn twist around a. When two simple closed curves a and b intersect in one point the associated Dehn twists satisfy the braid relation DaDbDa = DbDaDb, and if they do not intersect, the corresponding Dehn twists commute DaDb = DbDa. Thus a chain of n interlocking simple closed curves a1,...,an on some surface defines a map from the braid group Bn+1 on n+1 strands into the mapping class group of a subsurface Σ containing the union of theses curves; these mapping classes fix the boundary of Σ point-wise. The smallest such subsurface Σ is a neighbourhood of the union of the curves. When n = 2g + 1 is odd, this is Σg,2, and when n = 2g is even, this is Σg,1. Thus we have homomorphisms of groups φ : B2g+2 −→ Γg,2 and φ : B2g+1 −→ Γg,1. (1.1) These are injections by a theorem of Birman and Hilden [BH1], [BH2]. 1 a a 1 3 a2g+1 a2 a4 a2g Figure 1: The standard geometric embedding φ : B2g+2 → Γg,2. Wajnryb [W1] calls such embeddings that send the standard generators of the braid group to Dehn twists geometric. He asks in [W2] whether there are non- geometric embeddings. The first example of such a non-geometric embedding was given by Szepietowski [S]. Our first goal in this paper is to produce many more such non-geometric embeddings and show that they are ubiquitous. We construct these in section 2 and prove in section 3 that they are non-geometric. Motivated by a conjecture of Harer and following some ideas of F.R. Cohen [C], in [SoT] and [SeT] it was shown that the geometric embedding φ induces the trivial map in stable homology, that is the map in homology is zero in positive degrees as long as the genus of the underlying surface is large enough relative to the degree. Our second goal here is to show that this is also the case for the non-geometric embeddings constructed in section 2. (For one of these maps, this answers a question left open in [SoT].) To this purpose we show in section 4 how our embeddings from section 2 induce maps of algebras over an E2-operad, and deduce in section 5 that all of them induce the trivial map in stable homology. Furthermore, while it may be expected that all maps from a braid group to the mapping class group of an orientable surface will induce the trivial map on stable homology, we show that this is not true for embeddings of a braid group into the mapping class group of a non-orientable surface by explicitly computing the image of one such embedding in stable homology. Finally, in section 6, we analyse the induced maps in unstable homology. Only partial results are obtained here. In particular, it remains an open question whether φ induces the trivial map in unstable homology for field coefficients. Acknowledgement. We would like to thank Blazej Szepietowski for sending his paper and Mustafa Korkmaz for e-mail correspondence. 2 Non-geometric embeddings We will construct various injections of braid groups into mapping class groups. All embeddings that we know of, geometric or not, initially start with the stan- dard identification of the braid groups with mapping class groups. The pure 2 g braid group on g strands is the mapping class group Γ0,1 of a disk with g marked points, and the pure ribbon braid group is the mapping class group g g Γ0,g+1 ≃ Z ×Γ0,1 of a disk with g holes the boundary of which are parametrised. The factors of Z correspond to the Dehn twists around the boundary circles of the holes. Similarly, the braid group Bg can be identified with the mapping (g) class group Γ0,1 of a disk with g punctures (or g unordered marked points), and the ribbon braid group Z ≀ Bg with the mapping class group Γ0,(g),1 of the disk with g unordered holes for which the underlying diffeomorphisms may in a parametrisation preserving way interchange the boundaries of the holes and fix the outer boundary curve point-wise. Note that the resulting inclusion (γ : Bg ֒→ Z ≀ Bg ≃ Γ0,(g),1 (2.1 maps the standard generator that interchanges the i-th and (i + 1)-st strand in the braid group to half of the Dehn twists around a simple closed curve enclosing the i-th and (i + 1)-st holes followed by a half Dehn twist around each of these holes in the opposite direction; see Figure 2. σ Figure 2: Image of a generator σ ∈ Bg under γ. Clearly, any genus zero subsurface Σ0,g+1 of a surface Σ defines for us a map from the pure ribbon braid group Γ0,g+1 into the mapping class group of Σ. Not all such maps, however, can be extended to the ribbon braid group Γ0,(g),1. Below we explore various constructions of surfaces from Σ0,g+1 that allow such an extension. 2.1 Mirror construction. Our first example of a non-geometric embedding was also considered in [SoT]. We double the disk with g holes by reflecting it in a plane containing the boundary circles of the holes to obtain an oriented surface Σg−1,2 as indicated in Figure 3, and extend diffeomorphisms of the disk with holes to Σg−1,2 by reflection in the plane. This defines a map on mapping class groups R :Γ0,(g),1 −→ Γg−1,2. 3 Precomposing R with the natural inclusion Bg ⊂ Γ0,(g),1 defines the map γ R Bg −→ Γ0,(g),1 −→ Γg−1,2. (2.2) a2 a1 a3 ag Figure 3: Mirror construction R and subgroup ha1,...,agi ⊂ π1(Σg−1,2). Lemma 2.1. The composition R ◦ γ is an injection. Proof. We are going to detect the elements in the image by their action on the fundamental group of the underlying surface. Recall, the action of the braid group on the fundamental group of the disk with g holes defines Artin’s inclusion [A] of the braid group into the automorphism group of π1(Σ0,g+1) = Fg = ha1,...,agi, a free group of rank g: .(A : Bg ֒→ Aut(Fg The fundamental group of Σ0,g+1 in turn injects into the fundamental group of Σg−1,2. Indeed, each standard generator of Fg = ha1,...,agi is mapped to the corresponding generator of π1(Σg−1,2) which is the free group F2g−1 = ha1, b1,...,bg−1, agi of rank 2g − 1; see Figure 3. The action of diffeomorphisms on the fundamental group of the surface in- duces a group homomorphisms Γg−1,2 → Aut(F2g−1), and by restricting to the subgroup Fg = ha1,...,agi a map to the set Hom(Fg,F2g−1). We follow this map by the projection map to the monoid Hom(Fg,Fg), induced from the group homomorphisms F2g−1 → Fg that maps each ai to itself and each bi to the iden- tity element. We have the following commutative diagram; note that the right vertical arrow is only a map of sets. 4 R◦γ Bg −−−−→ Γg−1,2 A y y Aut(Fg) −−−−→ Hom(Fg,Fg). As A and the bottom horizontal map are injective this shows that R ◦ γ is injective. 2.2 Szepietowski’s construction. We next recall the construction from [S]. Starting with Σ0,g+1, the disk with g holes, we glue to the boundary of each hole a M¨obius band N1,1 so that the re- sulting surface is a non-orientable surface Ng,1 of genus g with one parametrised boundary component. Any diffeomorphisms f : Σ0,g+1 → Σ0,g+1 which fixes the boundary pointwise can be extended across the boundary by the identity of N1,1 ⊔ . ⊔ N1,1 and gives thus a diffeomorphism of Ng,1. This defines (via γ) a homomorphism from the pure braid group to the mapping class group of Ng,1. This latter homomorphism can be extended to a map from the whole braid group as follows: extend a diffeomorphism f which permutes the g inner boundaries by the corresponding permutation diffeomorphism of N1,1⊔. .⊔N1,1. This defines ϕ : Bg −→ Ng,1, (2.3) where Ng,1 denotes the mapping class group of Ng,1. By a result of Birman and Chillingworth [BC1], [BC2], extended to surfaces with boundary in [S], the lift of diffeomorphisms of a non-orientable surface to its double cover induces an injection of mapping class groups: .L : Ng,1 ֒→ Γg−1,2 Lemma 2.2. The composition L ◦ ϕ is an injection. Proof. We give an alternative proof to the one in [S].