Signatures of Topological Branched Covers

Christian Geske, Alexandra Kjuchukova*, Julius L. Shaneson

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Let X4 and Y4 be smooth manifolds and f : X → Y a branched cover with branching set B. Classically, if B is smoothly embedded in Y, the signature (X) can be computed from data about Y, B and the local degrees of f. When f is an irregular dihedral cover and B Y smoothly embedded away from a cone singularity whose link is K, the second author gave a formula for the contribution (K) to (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 f lat, away from the possible singularity. Owing to the presence of points on B which are not locally f lat, X in this setting is a stratified pseudomanifold, and we use the intersection homology signature of X, IH (X). For any knot K whose determinant is not ±1, a homotopy ribbon obstruction is derived from (K), providing a new technique to potentially detect slice knots that are not ribbon.

Original languageEnglish (US)
Pages (from-to)4605-4624
Number of pages20
JournalInternational Mathematics Research Notices
Volume2021
Issue number6
DOIs
StatePublished - Mar 1 2021

ASJC Scopus subject areas

  • Mathematics(all)

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