On finite symmetries of simply connected four-manifolds

TL;DR

This paper constructs infinitely many smooth structures and free finite group actions on certain simply connected four-manifolds, providing counterexamples to the 4D Rosenberg Conjecture and analyzing Yamabe invariant behavior.

Contribution

It introduces new smooth structures and free group actions on specific four-manifolds, challenging existing conjectures in 4D topology and geometry.

Findings

Existence of infinitely many inequivalent smooth structures on certain four-manifolds.

Construction of infinitely many non-equivalent smooth free actions of finite groups.

Identification of counterexamples to the 4D Rosenberg Conjecture.

Abstract

For most positive integer pairs $(a,b)$, the topological space $#a{\mathbb C \mathbb P}^2#b{\bar{\mathbb C \mathbb P^2}}$ is shown to admit infinitely many inequivalent smooth structures which dissolve upon performing a single connected sum with $S^2\times S^2$. This is then used to construct infinitely many non-equivalent smooth free actions of suitable finite groups on the connected sum $#a{\mathbb C \mathbb P}^2#b{\bar{\mathbb C \mathbb P^2}}$. We then investigate the behavior of the sign of the Yamabe invariant for the resulting finite covers, and observe that these constructions provide many new counter-examples to the $4$-dimensional Rosenberg Conjecture.