Exotic G_2-manifolds

TL;DR

This paper presents the first known examples of 7-dimensional G_2-holonomy manifolds that are homeomorphic but not diffeomorphic, using twisted connected sum constructions on Fano 3-folds, distinguished by a new invariant.

Contribution

It provides the first explicit examples of homeomorphic but not diffeomorphic G_2-manifolds and Ricci-flat manifolds, utilizing a novel invariant for smooth structure distinction.

Findings

First examples of G_2-manifolds with different smooth structures

Construction method using twisted connected sum on Fano 3-folds

Distinction of smooth structures via generalized Eells-Kuiper invariant

Abstract

We exhibit the first examples of closed 7-dimensional Riemannian manifolds with holonomy G_2 that are homeomorphic but not diffeomorphic. These are also the first examples of closed Ricci-flat manifolds that are homeomorphic but not diffeomorphic. The examples are generated by applying the twisted connected sum construction to Fano 3-folds of Picard rank 1 and 2. The smooth structures are distinguished by the generalised Eells-Kuiper invariant introduced by the authors in arXiv:1406.2226.