On the homotopy of the stable mapping class group

1997 ◽  
Vol 130 (2) ◽  
pp. 257-275 ◽  
Author(s):  
Ulrike Tillmann
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Stable Mapping

A splitting for the stable mapping class group

1999 ◽  
Vol 127 (1) ◽  
pp. 55-65 ◽  
Author(s):  
ULRIKE TILLMANN
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Loop Spaces ◽  
Classifying Space ◽  
Stable Homotopy Groups ◽  
Stable Mapping ◽  
Infinite Loop Spaces ◽  
Infinite Loop ◽  
Plus Construction

The main result of [15] is that the classifying space of the stable mapping class group after plus construction BΓ+∞ is an infinite loop space. This result is used to show that, localized away from two, a connected component of the stable homotopy groups of spheres QS0 splits off BΓ+∞. The splitting is a splitting of infinite loop spaces. It follows immediately that the homology with coefficients in ℤ[½] of the infinite symmetric group is a direct summand of the homology of the stable mapping class group.

scholarly journals Mod p homology of the stable mapping class group

Topology ◽  
2004 ◽  
Vol 43 (5) ◽  
pp. 1105-1132 ◽  
Author(s):  
Søren Galatius
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Stable Mapping ◽  
Mod P

The stable mapping class group and Q(ℂP∞+)

2001 ◽  
Vol 145 (3) ◽  
pp. 509-544 ◽  
Author(s):  
Ib Madsen ◽  
Ulrike Tillmann
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Stable Mapping

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

scholarly journals The local-to-global property for Morse quasi-geodesics

2021 ◽  
Author(s):  
Jacob Russell ◽  
Davide Spriano ◽  
Hung Cong Tran
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Type Theorem ◽  
Hyperbolic Groups ◽  
Global Property ◽  
Mapping Class ◽  
Relatively Hyperbolic Groups ◽  
Convex Cocompact ◽  
Relatively Hyperbolic ◽  
Translation Length

AbstractWe show the mapping class group, $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

Sign in / Sign up

Export Citation Format

Share Document