Singular branched covers of four-manifolds

Abstract

Consider a dihedral cover $f: Y\to X$ with $X$ and $Y$ four-manifolds and $f$ branched along an oriented surface embedded in $X$ with isolated cone singularities. We prove that only a slice knot can arise as the unique singularity on an irregular dihedral cover $f: Y\to S^4$ if $Y$ is homotopy equivalent to $\mathbb{CP}^2$ and construct an explicit infinite family of such covers with $Y$ diffeomorphic to $\mathbb{CP}^2$. An obstruction to a knot being homotopically ribbon arises in this setting, and we describe a class of potential counter-examples to the Slice-Ribbon Conjecture. Our tools include lifting a trisection of a singularly embedded surface in a four-manifold $X$ to obtain a trisection of the corresponding irregular dihedral branched cover of $X$, when such a cover exists. We also develop a combinatorial procedure to compute, using a formula by the second author, the contribution to the signature of the covering manifold which results from the presence of a singularity on the branching set.