Knot concordance and homology cobordism

Tim D. Cochran, Bridget D. Franklin, Matthew Hedden, Peter D. Horn

Research output: Contribution to journalArticlepeer-review

18 Scopus citations

Abstract

We consider the question: "If the zero-framed surgeries on two oriented knots in S3 are ℤ-homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?" We show that this question has a negative answer in the smooth category, even for topologically slice knots. To show this we first prove that the zero-framed surgery on K is ℤ-homology cobordant to the zero-framed surgery on many of its winding number one satellites P(K). Then we prove that in many cases the τ and s-invariants of K and P(K) differ. Consequently neither τ nor s is an invariant of the smooth homology cobordism class of the zero-framed surgery. We also show that a natural rational version of this question has a negative answer in both the opological and smooth categories by proving similar results for K and its (p, 1)-cables.

Original languageEnglish (US)
Pages (from-to)2193-2208
Number of pages16
JournalProceedings of the American Mathematical Society
Volume141
Issue number6
DOIs
StatePublished - 2013

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Fingerprint Dive into the research topics of 'Knot concordance and homology cobordism'. Together they form a unique fingerprint.

Cite this