Homological stability for the mapping class groups of non-orientable surfaces

Abstract

We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of non-orientable surfaces, up to homology isomorphism, is the infinite loop space of a Thom spectrum build from the canonical bundle over the Grassmannians of 2-planes in R^{n+2}. In particular, we show that the stable rational cohomology is a polynomial algebra on generators in degrees 4i--this is the non-oriented analogue of the Mumford conjecture.

References in corpus (4)

• Stabilization for mapping class groups of 3-manifolds
• Erratum to: Homology stability for outer automorphism groups of free groups
• Generating mapping class groups of nonorientable surfaces with boundary
• The homotopy type of the cobordism category

Cited by in corpus (28)

• Stabilization for mapping class groups of 3-manifolds
• Homological stability for moduli spaces of high dimensional manifolds. I
• Monoids of moduli spaces of manifolds
• Homological stability for mapping class groups of surfaces
• Crossed simplicial groups and structured surfaces
• Homological stability for automorphism groups
• Tethers and homology stability for surfaces
• Homological stability of topological moduli spaces
• The homology of the Brauer algebras
• A short exposition of the Madsen-Weiss theorem
• A twisted homology fibration criterion and the twisted group-completion theorem
• A presentation for the mapping class group of a nonorientable surface
• The homotopy type of the cobordism category
• The first homology group of the mapping class group of a nonorientable surface with twisted coefficients
• On the divisibility of characteristic classes of non-oriented surface bundles
• Torelli buildings and their automorphisms
• Lectures on invertible field theories
• The homology of a Temperley-Lieb algebra on an odd number of strands
• Advanced topics in gauge theory: mathematics and physics of Higgs bundles
• Singular cobordism categories
• The homology of the stable non-orientable mapping class group
• Stability phenomena in the topology of moduli spaces
• Lengths of geodesics on non-orientable hyperbolic surfaces
• Homological Stability for Spaces of Subsurfaces with Tangential Structure
• Disordered arcs and Harer stability
• Mapping class groups and function spaces: a survey
• Buildings, Group Homology and Lattices
• Homological stability of non-orientable mapping class groups with marked points