Signatures of Topological Branched Covers

Let $X^4$ and $Y^4$ be smooth manifolds and $f: X\to Y$ a branched cover with branching set $B$. Classically, if $B$ is smoothly embedded in $Y$, the signature $\sigma (X)$ can be computed from data about $Y$, $B$ and the local degrees of $f$. When $f$ is an irregular dihedral cover and $B\subset Y$ smoothly embedded away from a cone singularity whose link is $K$, the second author gave a formula for the contribution $\Xi (K)$ to $\sigma (X)$ resulting from the non-smooth point. We extend the above results to the case where $Y$ is a topological four-manifold and $B$ is locally flat, away from the possible singularity. Owing to the presence of points on $B$ which are not locally flat, $X$ in this setting is a stratified pseudomanifold, and we use the intersection homology signature of $X$, $\sigma _{IH}(X)$. For any knot $K$ whose determinant is not $\pm 1$, a homotopy ribbon obstruction is derived from $\Xi (K)$, providing a new technique to potentially detect slice knots that are not ribbon.

Curvature, cones and characteristic numbers

We study Einstein metrics on smooth compact 4-manifolds with an edge-cone singularity of specified cone angle along an embedded 2-manifold. To do so, we first derive modified versions of the Gauss–Bonnet and signature theorems for arbitrary Riemannian 4-manifolds with edge-cone singularities, and then show that these yield non-trivial obstructions in the Einstein case. We then use these integral formulæ to obtain interesting information regarding gravitational instantons which arise as limits of such edge-cone manifolds.