Exotic non-orientable four-manifolds with prescribed fundamental group

TL;DR

This paper constructs non-orientable four-manifolds with any finitely presented fundamental group (having an index two subgroup), exhibiting exotic smooth structures obtained via Gluck twists, and analyzes their stabilization properties.

Contribution

It demonstrates the realization of arbitrary finitely presented groups as fundamental groups of exotic non-orientable four-manifolds with specific smooth structure properties.

Findings

Fundamental groups with index two subgroups are realizable as non-orientable four-manifolds.

Exotic smooth structures are constructed via Gluck twists.

Stability of smooth structures under connected sums with certain 4-manifolds.

Abstract

We show that any finitely presented group with an index two subgroup is realized as the fundamental group of a closed smooth non-orientable four-manifold that admits an exotic smooth structure, which is obtained by performing a Gluck twist. The orientation 2-covers of these four-manifolds are diffeomorphic. These two smooth structures remain inequivalent after adding arbitrarily many copies of the product of a pair of 2-spheres and stabilize after adding a single copy of the complex projective plane.