Abelian quotients of mapping class groups of highly connected manifolds

2014/07/17 by Søren Galatius, Galatius, Soren, Oscar Randal‐Williams +1 · 1 citation
Mathematics · #55N22 #55P47 #55Q10 #57R15 #57S05 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology

paper · doi:10.48550/arxiv.1407.4680

openalex publication_date 2014/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We compute the abelianisations of the mapping class groups of the manifolds W_g²ⁿ = g(Sⁿ × Sⁿ) for n ≥ 3 and g ≥ 5. The answer is a direct sum of two parts. The first part arises from the action of the mapping class group on the middle homology, and takes values in the abelianisation of the automorphism group of the middle homology. The second part arises from bordism classes of mapping cylinders and takes values in the quotient of the stable homotopy groups of spheres by a certain subgroup which in many cases agrees with the image of the stable J-homomorphism. We relate its calculation to a purely homotopy theoretic problem.

Citations

• Homological stability and stable moduli of flat manifold bundles

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