Quotients of $S^2\times{S^2}$

TL;DR

This paper classifies certain 4-manifolds with universal cover S^2×S^2, focusing on their fundamental groups and homeomorphism types, and explores smooth structures and potential quotients.

Contribution

It determines the number of homeomorphism types for manifolds with specific fundamental groups and proposes a candidate for a non-homeomorphic smooth quotient.

Findings

Four homeomorphism types for π ≅ Z/4.

Three homotopy types for π ≅ Z/2 × Z/2.

Between 6 and 24 homeomorphism types in the latter case.

Abstract

We consider closed topological 4-manifolds $M$ with universal cover ${S^2\times{S^2}}$ and Euler characteristic $\chi(M) = 1$. All such manifolds with $\pi=\pi_1(M)\cong {\mathbb Z}/4$ are homotopy equivalent. In this case, we show that there are four homeomorphism types, and propose a candidate for a smooth example which is not homeomorphic to the geometric quotient. If $\pi\cong {\mathbb Z}/2 \times {\mathbb Z}/2$, we show that there are three homotopy types (and between 6 and 24 homeomorphism types).