## Knot concordance and homology cobordism

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- by Tim D. Cochran, Bridget D. Franklin, Matthew Hedden and Peter D. Horn PDF
- Proc. Amer. Math. Soc.
**141**(2013), 2193-2208 Request permission

## 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.## References

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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
- MR Author ID: 769768
- 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
- MR Author ID: 855878
- Email: pdhorn@math.columbia.edu, pdhorn@syr.edu
- 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
- © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**141**(2013), 2193-2208 - MSC (2010): Primary 57N70, 57M25
- DOI: https://doi.org/10.1090/S0002-9939-2013-11471-1
- MathSciNet review: 3034445