Dihedral branched covers of four-manifolds

Abstract

Given a closed oriented PL four-manifold X and a closed surface B embedded in X with isolated cone singularities, we give a formula for the signature of an irregular dihedral cover of X branched along B. For X simply-connected, we deduce a necessary condition on the intersection form of a simply-connected irregular dihedral branched cover of (X,B). When the singularities on B are two-bridge slice, we prove that the necessary condition on the intersection form of the cover is sharp. For X a simply-connected PL four-manifold with non-zero second Betti number, we construct infinite families of simply-connected PL manifolds which are irregular dihedral branched coverings of X. Given two four-manifolds X and Y whose intersection forms are odd, we obtain a necessary and sufficient condition for Y to be homeomorphic to an irregular dihedral p-fold cover of X, branched over a surface with a two-bridge slice singularity.