Abelian quotients of mapping class groups of highly connected manifolds

TL;DR

This paper computes the abelian quotients of mapping class groups for highly connected manifolds, revealing their structure through homology actions and bordism classes, and relates the problem to stable homotopy theory.

Contribution

It provides explicit calculations of the abelianisations of mapping class groups for certain manifolds, connecting algebraic and homotopy theoretic methods.

Findings

Abelianisation splits into homology action and bordism class parts.

The homology part relates to automorphism groups of middle homology.

The bordism part involves stable homotopy groups and the stable J-homomorphism.

Abstract

We compute the abelianisations of the mapping class groups of the manifolds $W_g^{2n} = g(S^n \times S^n)$ for $n \geq 3$ and $g \geq 5$. 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.