Some examples of small irreducible exotic 4-manifolds with free abelian fundamental group

TL;DR

This paper constructs new examples of small, irreducible 4-manifolds with free abelian fundamental groups, revealing their exotic smooth structures and how they become standard after stabilization.

Contribution

It introduces an explicit method to compute equivariant intersection forms of 4-manifolds obtained through torus surgeries and presents a new stabilization result for exotic smooth structures.

Findings

Examples of non-diffeomorphic 4-manifolds with free abelian fundamental groups of rank less than three.

Explicit computation method for equivariant intersection forms after torus surgeries.

Stability of exotic smooth structures under connected sum with S^2×S^2.

Abstract

We produce examples of pairwise non-diffeomorphic closed irreducible 4-manifolds with non-trivial free abelian fundamental group of rank less than three and small Euler characteristic. These exotic smooth structures become standard after taking a connected sum with a single copy of $S^2\times S^2$. The contributions of this paper include an explicit mechanism to computate the equivariant intersection form of 4-manifolds that are obtained via torus surgeries and a new stabilization result concerning exotic smooth structures with arbitrary fundamental group.