On smooth structures over $4$-manifolds with fundamental group of even order

TL;DR

This paper investigates the existence and abundance of smooth structures on certain 4-manifolds with specific fundamental groups, extending previous results to new classes of manifolds.

Contribution

It establishes conditions under which 4-manifolds with particular fundamental groups have either no or infinitely many smooth structures, and constructs new examples with specified properties.

Findings

Manifolds with fundamental group Z_{4k} and large b_2 have either none or infinitely many smooth structures.

Constructs infinitely many non-diffeomorphic smooth structures on manifolds with fundamental group Z_2×G and signature zero.

Extends prior results by Baykur-Stipsicz-Szabó to broader classes of 4-manifolds.

Abstract

We show that any topological, closed, oriented, non-spin $4$-manifold with fundamental group $\mathbb{Z}_{4k}$ and $\min(b_2^+, b_2^-)\geq 15$, has either none or infinitely many distinct smooth structures. Furthermore, we construct infinitely many non-diffeomorphic, irreducible, smooth structures on manifolds with signature zero, $b_2^+$ even and fundamental group $\mathbb{Z}_2\times G$, for any finite group $G$. This extends the results of Baykur-Stipsicz-Szab\'o.