### Abstract

We consider the question: "If the zero-framed surgeries on two oriented knots in S^{3} 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 language | English (US) |
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Pages (from-to) | 2193-2208 |

Number of pages | 16 |

Journal | Proceedings of the American Mathematical Society |

Volume | 141 |

Issue number | 6 |

DOIs | |

State | Published - Apr 2 2013 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Proceedings of the American Mathematical Society*,

*141*(6), 2193-2208. https://doi.org/10.1090/S0002-9939-2013-11471-1