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Journal Article

Signatures of topological branched covers


Kjuchukova,  Alexandra
Max Planck Institute for Mathematics, Max Planck Society;

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Geske, C., Kjuchukova, A., & Shaneson, J. L. (in press). Signatures of topological branched covers. International Mathematics Research Notices, Published Online - Print pending. doi:10.1093/imrn/rnaa184.

Cite as: http://hdl.handle.net/21.11116/0000-0007-745C-0
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.