Stripping and splitting decorated mapping class groups

Bödigheimer, Carl-Friedrich; Tillmann, Ulrike

doi:10.1007/978-3-0348-8312-2_6

Cited by 23 publications

(55 citation statements)

References 12 publications

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“…Our second result is the following splitting theorem for the classifying spaces of these groups. This is analogous to Theorem 3.1 in [2] where the orientable case was considered.…”

Section: Introductionsupporting

confidence: 63%

See 1 more Smart Citation

Homological stability of non-orientable mapping class groups with marked points

Hanbury

2008

Proc. Amer. Math. Soc.

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“…Our second result is the following splitting theorem for the classifying spaces of these groups. This is analogous to Theorem 3.1 in [2] where the orientable case was considered.…”

Section: Introductionsupporting

confidence: 63%

“…is trivial for g in this range (this is proved in the same way as Lemma 3.3 in [2]), so the coefficients are not twisted. (2) k then, for g ≥ 4q + 3, we can apply Theorem 1.5 and deduce that the map on the E 2 -term is an isomorphism for…”

Section: Changing the Number Of Marked Pointsmentioning

confidence: 72%

Homological stability of non-orientable mapping class groups with marked points

Hanbury

2008

Proc. Amer. Math. Soc.

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“…, ∂ k . As a consequence of Harer-Ivanov stability [12], [13], it was proved in [5] that the map β k → Γ g+k,(k),1 factors in homology in degrees * < (g + k)/2 through Σ k . The right most vertical map is a homology isomorphism in these degrees, and hence the claim follows as the image of π : β k → Σ k in homology contains only 2-torsion, cf.…”

Section: Surface Category and The Functor φmentioning

confidence: 99%

“…All trees that occur have vertices that are at most trivalent. By an inductive argument, it is enough to consider the tree T in Figure 4 with five vertices, 1, 2, 3, 4, 5, and edges (1, 2), (2, 4)(3, 4), (4,5). The corresponding generators are mapped to four elements …”

Section: Definition Of Tile Categoriesmentioning

confidence: 99%

Braids, mapping class groups, and categorical delooping

Song

Tillmann

2007

Math. Ann.

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Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic map φ : β 2g → Γ g,1 from the braid group to the mapping class group. We prove here that this map is trivial in homology with any trivial coefficients in degrees less than g/2. In particular this proves an old conjecture of J. Harer. The main tool is categorical delooping in the spirit of [25]. By extending the homomorphism to a functor of monoidal 2-categories, φ is seen to induce a map of double loop spaces on the plus construction of the classifying spaces. Any such map is null-homotopic. In an appendix we show that geometrically defined homomorphisms from the braid group to the mapping class group behave similarly in stable homology.

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“…The following is a result of Bödigheimer and Tillmann in [2] and it follows directly from the preceeding corollary.…”

Section: Corollary 14 the Chain Complexmentioning

confidence: 64%

The unstable integral homology of the mapping class groups of a surface with boundary

Godin

2006

Math. Ann.

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Abstract. We construct a graph complex calculating the integral homology of the bordered mapping class groups. We compute the homology of the bordered mapping class groups of the surfaces S 1,1 , S 1,2 and S 2,1 . Using the circle action on this graph complex, we build a double complex and a spectral sequence converging to the homology of the unbordered mapping class groups. We compute the homology of the punctured mapping class groups associated to the surfaces S 1,1 and S 2,1 . Finally, we use Miller's operad to get the first Kudo-Araki and Browder operations on our graph complex. We also consider an unstable version of the higher Kudo-Araki-Dyer-Lashoff operations.

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Stripping and splitting decorated mapping class groups

Cited by 23 publications

References 12 publications

Homological stability of non-orientable mapping class groups with marked points

Homological stability of non-orientable mapping class groups with marked points

Braids, mapping class groups, and categorical delooping

The unstable integral homology of the mapping class groups of a surface with boundary

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