Smooth Structures on the product of 3-connected 8-manifolds with spheres

TL;DR

This paper investigates the smooth structures on products of 3-connected 8-manifolds with spheres, computing their concordance inertia groups and classifying all such manifolds up to concordance and diffeomorphism.

Contribution

It provides the first detailed computation of the concordance inertia group for these product manifolds and offers a classification of all homeomorphic smooth structures up to concordance and diffeomorphism.

Findings

Computed the concordance inertia group for M×S^k, 1≤k≤14

Classified all smooth manifolds homeomorphic to M×S^k for 1≤k≤10

Provided a diffeomorphism classification for M×S^1 when H^4(M;Z)=Z

Abstract

Let $M$ be a closed, 3-connected, 8-dimensional smooth manifold. In this paper, we compute the concordance inertia group of the product manifold $M\times\mathbb{S}^k$ for $1\leq k\leq 14$ and classify all smooth manifolds homeomorphic to $M\times\mathbb{S}^k,$ up to concordance for $1\leq k\leq 10.$ Moreover, we provide a diffeomorphism classification of smooth manifolds homeomorphic to $M\times\mathbb{S}^1,$ where $H^4(M;\mathbb{Z})=\mathbb{Z}.$