Branched coverings of $CP^2$ and other basic 4-manifolds

TL;DR

This paper establishes criteria based on Betti numbers and intersection forms for when a 4-manifold can be represented as a branched cover of fundamental 4-manifolds like $CP^2$ and $S^2 imes S^2$, aiding classification efforts.

Contribution

It provides necessary and sufficient conditions for 4-manifolds to be branched covers of key 4-manifolds, linking topological invariants to covering properties.

Findings

Criteria expressed via Betti numbers and intersection forms.

Conditions applicable to multiple fundamental 4-manifolds.

Facilitates classification of 4-manifolds as branched covers.

Abstract

We give necessary and sufficient conditions for a 4-manifold to be a branched covering of $CP^2$, $S^2\times S^2$, $S^2 \mathbin{\tilde\times} S^2$ and $S^3 \times S^1$, which are expressed in terms of the Betti numbers and the intersection form of the 4-manifold.