The Stable Mapping Class Group and Stable Homotopy Theory

2009 ◽  
pp. 283-308 ◽  
Author(s):  
Jørgen Ellegaard Andersen ◽  
Michael Weiss
Keyword(s):  
Mapping Class Group ◽  
Homotopy Theory ◽  
Class Group ◽  
Stable Homotopy ◽  
Mapping Class ◽  
Stable Homotopy Theory ◽  
Stable Mapping

scholarly journals On the Cohomology of the Mapping Class Group of the Punctured Projective Plane

2020 ◽  
Vol 71 (2) ◽  
pp. 539-555
Author(s):  
Miguel A Maldonado ◽  
Miguel A Xicoténcatl
Keyword(s):  
Exact Sequence ◽  
Configuration Space ◽  
Mapping Class Group ◽  
Homotopy Theory ◽  
Class Group ◽  
Orientable Surface ◽  
Configuration Spaces ◽  
Mapping Class ◽  
Serre Spectral Sequence ◽  
Classical Homotopy

Abstract The mapping class group $\Gamma ^k(N_g)$ of a non-orientable surface with punctures is studied via classical homotopy theory of configuration spaces. In particular, we obtain a non-orientable version of the Birman exact sequence. In the case of ${\mathbb{R}} \textrm{P}^2$, we analyze the Serre spectral sequence of a fiber bundle $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k \to X_k \to BSO(3)$ where $X_k$ is a $K(\Gamma ^k({\mathbb{R}} \textrm{P}^2),1)$ and $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k$ denotes the configuration space of unordered $k$-tuples of distinct points in ${\mathbb{R}} \textrm{P}^2$. As a consequence, we express the mod-2 cohomology of $\Gamma ^k({\mathbb{R}} \textrm{P}^2)$ in terms of that of $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k$.

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
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