A short exposition of the Madsen-Weiss theorem

2011/03/27 by Allen Hatcher, Hatcher, Allen · 3 citations
Mathematics · #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1103.5223

openalex publication_date 2011/03/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This is an exposition of a proof of the Madsen-Weiss Theorem, which asserts that the homology of mapping class groups of surfaces, in a stable dimension range, is isomorphic to the homology of a certain infinite loopspace that arises naturally when one applies the "scanning method". The proof given here utilizes simplifications introduced by Galatius and Randal-Williams.

Cited by

• The Barratt--Priddy--Quillen theorem via scanning methods
• A local to global argument on low dimensional manifolds
• Normal invariant of nearby Lagrangians via twisted derivative

Related

• Strings and the stable cohomology of mapping class groups
• Surfaces in a background space and the homology of mapping class groups
• Homological stability of non-orientable mapping class groups with marked points
• Mod p homology of the stable mapping class group
• Improved homological stability for the mapping class group with integral or twisted coefficients