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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Knot concordance and homology cobordism


Authors: Tim D. Cochran, Bridget D. Franklin, Matthew Hedden and Peter D. Horn
Journal: Proc. Amer. Math. Soc. 141 (2013), 2193-2208
MSC (2010): Primary 57N70, 57M25
Published electronically: January 11, 2013
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the question: ``If the zero-framed surgeries on two oriented knots in $ S^3$ are $ \mathbb{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 $ \mathbb{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 $ \tau $ and $ s$-invariants of $ K$ and $ P(K)$ differ. Consequently neither $ \tau $ 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 topological and smooth categories by proving similar results for $ K$ and its $ (p,1)$-cables.


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Additional Information

Tim D. Cochran
Affiliation: Department of Mathematics, Rice University, Houston, Texas 77251
Email: cochran@math.rice.edu

Bridget D. Franklin
Affiliation: Department of Mathematics, Rice University, Houston, Texas 77251
Email: bridget.franklin@alumni.rice.edu

Matthew Hedden
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: mhedden@math.msu.edu

Peter D. Horn
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
Address at time of publication: Department of Mathematics, Syracuse University, Syracuse, New York 13244
Email: pdhorn@math.columbia.edu, pdhorn@syr.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11471-1
PII: S 0002-9939(2013)11471-1
Received by editor(s): November 22, 2010
Received by editor(s) in revised form: April 13, 2011, September 28, 2011, and September 30, 2011
Published electronically: January 11, 2013
Additional Notes: The first author was partially supported by National Science Foundation DMS-1006908
The second author was partially supported by Nettie S. Autry Fellowship
The third author was partially supported by NSF DMS-0906258
The fourth author was partially supported by NSF Postdoctoral Fellowship DMS-0902786
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.