On the mapping class groups of $#_r(S^p \times S^p)$ for $p = 3, 7$

TL;DR

This paper computes the structure of the mapping class groups and homotopy classes of orientation-preserving diffeomorphisms and homotopy equivalences for manifolds formed by connected sums of products of spheres in dimensions 6 and 14.

Contribution

It provides explicit calculations of the mapping class groups for manifolds M_r = #_r(S^p × S^p) with p=3,7, extending understanding of their diffeomorphism and homotopy class structures.

Findings

Calculated the group of isotopy classes of orientation-preserving diffeomorphisms for M_r.

Determined the group of homotopy classes of orientation-preserving homotopy equivalences.

Extended the classification results to specific high-dimensional manifolds.

Abstract

For M_r = #_r(S^p \times S^p), p=3, 7, we calculate the group of isotopy classes of orientation preserving diffeomorphisms of $M_r$ modulo isotopy classes with representatives which are the identity outside a 2p-disc and also the group of homotopy classes of orientation preserving homotopy equivalences of M_r.