scholarly journals The stable mapping class group of simply connected 4-manifolds

2008 ◽  
Vol 2008 (617) ◽  
Author(s):  
Jeffrey Giansiracusa
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Stable Mapping ◽  
Simply Connected

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

scholarly journals The action of the mapping class group on metrics of positive scalar curvature

2021 ◽  
Author(s):  
Georg Frenck
Keyword(s):  
Scalar Curvature ◽  
Mapping Class Group ◽  
Class Group ◽  
Index Theorem ◽  
Positive Scalar Curvature ◽  
Mapping Class ◽  
High Dimensional ◽  
Positive Scalar ◽  
Simply Connected

AbstractWe present a rigidity theorem for the action of the mapping class group $$\pi _0({\mathrm{Diff}}(M))$$ π 0 ( Diff ( M ) ) on the space $$\mathcal {R}^+(M)$$ R + ( M ) of metrics of positive scalar curvature for high dimensional manifolds M. This result is applicable to a great number of cases, for example to simply connected 6-manifolds and high dimensional spheres. Our proof is fairly direct, using results from parametrised Morse theory, the 2-index theorem and computations on certain metrics on the sphere. We also give a non-triviality criterion and a classification of the action for simply connected 7-dimensional $${\mathrm{Spin}}$$ Spin -manifolds.

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.

Sign in / Sign up

Export Citation Format

Share Document