TY - JOUR

T1 - Signatures of Topological Branched Covers

AU - Geske, Christian

AU - Kjuchukova, Alexandra

AU - Shaneson, Julius L.

N1 - Publisher Copyright:
© 2020 The Author(s) 2020. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com.

PY - 2021/3/1

Y1 - 2021/3/1

N2 - 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.

AB - 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.

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U2 - 10.1093/imrn/rnaa184

DO - 10.1093/imrn/rnaa184

M3 - Article

AN - SCOPUS:85109292666

SN - 1073-7928

VL - 2021

SP - 4605

EP - 4624

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 6

ER -