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Mathematics, Geometric Topology
Abstract:
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 {\it topological} four-manifold and $B$ is locally flat, away from the
possible singularity. Owing to the presence of non-locally-flat points on $B$,
$X$ 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.
