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

By Nathalie Wahl. 2006

72 citations recorded by the source corpus.

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 built from the canonical bundle over the Grassmannians of 2-planes in ℝn+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.

Subfields: Geometry and Topology, Mathematical Physics, Surgery

View Paper: https://www.semanticscholar.org/paper/ce83a4325b392a8fb7d2c2c34e76bddc1c743d55

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Citations in the corpus

Homological stability for mapping class groups of surfaces (2010, methodology): “The arguments given in this section are collected from the papers [13, 16, 20, 28, 30].”
Homology of the moduli spaces and mapping class groups of framed, r‐Spin and Pin surfaces (2010, background): “In §5.1 we also study the divisibility of certain characteristic classes ζi which were defined by Wahl in the integral cohomology of the moduli spaces of unoriented surfaces, when they are pulled back to the moduli spaces MPin ± (F ).”
Monoids of moduli spaces of manifolds (2009, background): “imit of the mapping class groups Γg,1 as g→∞. A similar result holds for non-orientable or spin surfaces, though the components of the relevant monoid are slightly more complicated. Corollary D (Wahl [Wah08] Theorem B). Let Ng be the mapping class group of a non-orientable surface of genus g. Then t”
Surfaces in a background space and the homology of mapping class groups (2006, background): “These are the “incoming and outgoing” boundary components of S. (S, ψ) is a morphism from (∂aS, ψ|∂aS ) to (∂bS, ψ|∂bS ) Composition in this category is given by union of surfaces along common, parameterized boundary components.”
Resolutions of moduli spaces and homological stability (2009, methodology): “The purpose of this appendix is to deduce information about the complexes we need from that available in [11, 25].”
The homotopy type of the cobordism category (2006, background): “homological stability theorems have been proved for surfaces with tangential structure in a number of other situations, which we now list. •N. Wahl considered stability for non-orientable surfaces in [Wah08]. Let Sg,n denote the connected sum of gcopies of R P2 with ndisks cut out, and consider the ”
Homological stability for moduli spaces of high dimensional manifolds (2012, result): “is constant, and θis spherical; (ii) 3i≤g−4 and Lis constant; HOMOLOGICAL STABILITY II 5 (iii) 3i≤g−4, g≥5, and either g≥7 or θis spherical. The situation is formally quite similar to both [HV04] and [Wah08], from which we learnt the following trick. Suppose one wishes to prove homological stability”
Homological stability for automorphism groups (2014, methodology): “For constant coefficients, this recovers the first part of Theorem A in [46], with almost the improved range of [39, 1.”