## Brunnian subgroups of mapping class groups and braid groups (2010)

### BibTeX

@MISC{Berrick10brunniansubgroups,

author = {A. J. Berrick and E. Hanbury and J. Wu},

title = {Brunnian subgroups of mapping class groups and braid groups},

year = {2010}

}

### OpenURL

### Abstract

In this paper we continue our study of the Delta-group structure on the braid groups and mapping class groups of a surface. We calculate the homotopy groups of these Delta-groups and prove some results about Brunnian braid groups and Brunnian mapping class groups. This is the second of a pair of papers on these structures. 1 Introduction and statement of results In this paper we study the sequences

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Citation Context ...S 2 ) is trivial. Theorem 4.5 of the same book shows that Γ (2) (S 2 ) ∼ = Σ2 and Γ (3) (S 2 ) ∼ = Σ3. (The fact that the map Γ (3) (S 2 ) → Σ3 is an isomorphism is also proved in Proposition 2.13 of =-=[10]-=-). It follows that Γ2 (S2) = Γ (2)(S2)/Σ2 and Γ (3)(S2) = Γ3 (S2)/Σ3 are both trivial. □ Proposition 3.8. The mapping class group Γ 1 (T) is isomorphic to SL2(Z). Proof. It is well-known that Γ(T) ∼ =... |

51 |
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Citation Context ...up of Diff(M) consisting of all diffeomorphisms that are isotopic to the identity. This can alternatively be described as the path-component of the identity in Diff(M). 6Theorem 3.4 (Corollary 1E in =-=[6]-=-). Let M be a closed surface. Then Diff0(M) is contractible unless M is S2 , T, RP 2 or K. In the exceptional cases we have Diff0(S2 ) ≃ SO(3) Diff0(T) ≃ T Diff0(RP 2 ) ≃ SO(3) Diff0(K) ≃ SO(2). Theor... |

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(Show Context)
Citation Context ...cket arrangements b t (x1, . . .,xt) such that each x belongs to some Ri and there is at least one x from each Ri, see [13]. Fat commutator subgroups are important in homotopy theory, see for example =-=[18]-=- where they are used to describe the homotopy groups of spheres and of suspensions ΣK(π, 1). Our calculations will be based on the following theorem. It can be obtained from Theorem 1.1 and Equation (... |

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(Show Context)
Citation Context ...ases π2(Confk(S 2 )) for k ≥ 3 and π2(Confk(RP 2 )) for k ≥ 2. The remaining cases follow since Conf1(M) ∼ = M for any M and S2 ≃ Conf2(S2 ) via the map z ↦→ (z, −z). □ Proposition 3.2 ([4], p.34 and =-=[9]-=-). P1(S 2 ) ∼ = π1(S 2 ) = 1, P2(S 2 ) = 1 and P3(S 2 ) ∼ = Z/2. Proposition 3.3 ([17], Section III). P1(RP 2 ) ∼ = π1(RP 2 ) ∼ = Z/2 and P2(RP 2 ) ∼ = Q8. Below, Diff0(M) denotes the subgroup of Diff... |

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Teichmüller theory for surfaces with boundary
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Citation Context ...urface. Then Diff0(M) is contractible unless M is S2 , T, RP 2 or K. In the exceptional cases we have Diff0(S2 ) ≃ SO(3) Diff0(T) ≃ T Diff0(RP 2 ) ≃ SO(3) Diff0(K) ≃ SO(2). Theorem 3.5 (Theorem 1D in =-=[7]-=-). If M is a surface with boundary then Diff0(M) is contractible. Theorem 3.6 (Corollary 4.5 in [3]). Let M be a surface with k ≥ 1 marked points. Then Diff0(M,mk) is contractible unless M = S 2 and k... |

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(Show Context)
Citation Context ...˜α(1) ∈ Diff(T, m0) is isotopic to the identity relative to the marked point. Thus ∂([α]) = [˜α(1)] = 1. □ Proposition 3.9 (First bit is either in [8] or [14] - need to check, Theorems 4.1 and 4.3 in =-=[12]-=-). Γ(RP 2 ) = 1, Γ 1 (RP 2 ) = Z/2 and Γ 2 (RP 2 ) = Z/2 ⊕ Z/2. 73.2 Identifying the kernels of the face maps Lemma 3.10. Let ⎧ ⎨ 1 if M ̸= S k ≥ ⎩ 2 , RP 2 , 4 if M = S2 , 3 if M = RP 2 . (i) For an... |

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Citation Context ..., as are the cases π2(Confk(S 2 )) for k ≥ 3 and π2(Confk(RP 2 )) for k ≥ 2. The remaining cases follow since Conf1(M) ∼ = M for any M and S2 ≃ Conf2(S2 ) via the map z ↦→ (z, −z). □ Proposition 3.2 (=-=[4]-=-, p.34 and [9]). P1(S 2 ) ∼ = π1(S 2 ) = 1, P2(S 2 ) = 1 and P3(S 2 ) ∼ = Z/2. Proposition 3.3 ([17], Section III). P1(RP 2 ) ∼ = π1(RP 2 ) ∼ = Z/2 and P2(RP 2 ) ∼ = Q8. Below, Diff0(M) denotes the su... |

9 |
Homotopie des complexes monöideaux, Seminaire Henri Cartan
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Citation Context ...alization. If G is a simplicial group then |G| is a topological group in the category of compactly generated spaces. . In particular, it has a classifying space B|G|. Jie: this is Theorem 5.6 (Moore, =-=[16]-=-). Let G be a simplicial group and let |G| be its geometric realization. Then πk(G) ∼ = πk(|G|) for each k ≥ 0. □ Here πk(G) denotes the homotopy groups of G, defined in terms of the Moore chain compl... |

6 | On symmetric commutator subgroups, braids, links and homotopy groups
- Li, Wu
(Show Context)
Citation Context ... is denoted [[R1, . . .,Rn]] 10and is the subgroup generated by all possible bracket arrangements b t (x1, . . .,xt) such that each x belongs to some Ri and there is at least one x from each Ri, see =-=[13]-=-. Fat commutator subgroups are important in homotopy theory, see for example [18] where they are used to describe the homotopy groups of spheres and of suspensions ΣK(π, 1). Our calculations will be b... |

5 | Configurations, braids and homotopy groups - Berrick, R, et al. |

4 |
Brunnian braids on surfaces
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(Show Context)
Citation Context ... we have π2(Conf1(S 2 )) ∼ = Z ∼ = π2(Conf2(S 2 )), π2(Confk(S 2 )) = 0 for k ≥ 3, π2(Conf1(RP 2 )) ∼ = Z, π2(Confk(RP 2 )) = 0 for k ≥ 2. Proof. The non-exceptional cases are covered by Lemma 3.4 in =-=[1]-=-, as are the cases π2(Confk(S 2 )) for k ≥ 3 and π2(Confk(RP 2 )) for k ≥ 2. The remaining cases follow since Conf1(M) ∼ = M for any M and S2 ≃ Conf2(S2 ) via the map z ↦→ (z, −z). □ Proposition 3.2 (... |

3 | Delta-structures on mapping class groups and braid groups
- Berrick, Hanbury, et al.
(Show Context)
Citation Context ... points in M. If M has boundary then we insist that the diffeomorphisms restrict to the identity on the boundary and if M is orientable, the diffeomorphisms must be orientation-preserving. In [2] and =-=[3]-=-, it was shown that each of these sequences forms a ∆-group i.e. there are face maps di : Pk+1(M) → Pk(M) and di : Γ k+1 (M) → Γ k (M) 1for each i = 0, . . .,k and these satisfy a certain, natural id... |

2 |
Lickorish: Homeomorphisms of non-orientable two-manifolds
- R
- 1963
(Show Context)
Citation Context ... to its original position. The diffeomorphism ˜α(1) ∈ Diff(T, m0) is isotopic to the identity relative to the marked point. Thus ∂([α]) = [˜α(1)] = 1. □ Proposition 3.9 (First bit is either in [8] or =-=[14]-=- - need to check, Theorems 4.1 and 4.3 in [12]). Γ(RP 2 ) = 1, Γ 1 (RP 2 ) = Z/2 and Γ 2 (RP 2 ) = Z/2 ⊕ Z/2. 73.2 Identifying the kernels of the face maps Lemma 3.10. Let ⎧ ⎨ 1 if M ̸= S k ≥ ⎩ 2 , R... |

2 |
Buskirk: Braid groups of compact 2-manifolds with elements of …nite order
- van
- 1966
(Show Context)
Citation Context ...llow since Conf1(M) ∼ = M for any M and S2 ≃ Conf2(S2 ) via the map z ↦→ (z, −z). □ Proposition 3.2 ([4], p.34 and [9]). P1(S 2 ) ∼ = π1(S 2 ) = 1, P2(S 2 ) = 1 and P3(S 2 ) ∼ = Z/2. Proposition 3.3 (=-=[17]-=-, Section III). P1(RP 2 ) ∼ = π1(RP 2 ) ∼ = Z/2 and P2(RP 2 ) ∼ = Q8. Below, Diff0(M) denotes the subgroup of Diff(M) consisting of all diffeomorphisms that are isotopic to the identity. This can alte... |