Smooth structures on four-manifolds with finite cyclic fundamental groups

TL;DR

This paper investigates smooth structures on certain four-manifolds with cyclic fundamental groups, showing most do not admit smooth structures or have infinitely many, and constructs diverse fake projective planes.

Contribution

It classifies smooth structures on four-manifolds with specific cyclic fundamental groups and constructs new examples of fake projective planes with varied fundamental groups.

Findings

Most four-manifolds with fundamental group Z_{4m+2} and odd intersection form have no smooth structures or infinitely many.

Constructed infinite families of non-complex fake projective planes.

Identified exceptions where smooth structures may not exist or be infinite.

Abstract

For each nonnegative integer m we show that any closed, oriented topological four-manifold with fundamental group Z_{4m+2} and odd intersection form, with possibly seven exceptions, either admits no smooth structure or admits infinitely many distinct smooth structures up to diffeomorphism. Moreover, we construct infinite families of non-complex irreducible fake projective planes with diverse fundamental groups.