Title: 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.