The classification of 2-connected 7-manifolds

TL;DR

This paper provides a comprehensive classification of closed smooth spin 2-connected 7-manifolds, extending previous results by generalizing the Eells-Kuiper invariant and analyzing the inertia group related to smooth structures.

Contribution

It introduces an extended Eells-Kuiper invariant applicable to all closed spin 7-manifolds and determines the inertia group based on cohomological data.

Findings

Extended Eells-Kuiper invariant for all spin 7-manifolds

Classification of smooth structures via the inertia group

Relation between p_M, torsion linking form, and smooth structures

Abstract

We present a classification theorem for closed smooth spin 2-connected 7-manifolds M. This builds on the almost-smooth classification from the first author's thesis. The main additional ingredient is an extension of the Eells-Kuiper invariant for any closed spin 7-manifold, regardless of whether the spin characteristic class p_M in the fourth integral cohomology of M is torsion. In addition we determine the inertia group of 2-connected M - equivalently the number of oriented smooth structures on the underlying topological manifold - in terms of p_M and the torsion linking form.