# Mapping class groups of highly connected $(4k+2)$-manifolds

## TL;DR

This paper computes the mapping class groups of certain highly connected manifolds, revealing their structure in relation to homology automorphisms and homotopy spheres, and explores their Torelli subgroup and related groups.

## Contribution

It provides explicit descriptions of the mapping class groups of connected sums of sphere products, including their Torelli subgroups and abelianizations, extending understanding of their algebraic structure.

## Key findings

- Computed the mapping class group in terms of automorphisms of middle homology.
- Identified the Torelli subgroup of these manifolds.
- Determined the abelianizations of the mapping class groups.

## Abstract

We compute the mapping class group of the manifolds $\sharp^g(S^{2k+1}\times S^{2k+1})$ for $k>0$ in terms of the automorphism group of the middle homology and the group of homotopy $(4k+3)$-spheres. We furthermore identify its Torelli subgroup, determine the abelianisations, and relate our results to the group of homotopy equivalences of these manifolds.