#### Vol. 272, No. 1, 2014

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Nonconcordant links with homology cobordant zero-framed surgery manifolds

### Jae Choon Cha and Mark Powell

Vol. 272 (2014), No. 1, 1–33
##### Abstract

We use topological surgery theory to give sufficient conditions for the zero-framed surgery manifold of a 3-component link to be homology cobordant to the zero-framed surgery on the Borromean rings (also known as the 3-torus) via a topological homology cobordism preserving the free homotopy classes of the meridians.

This enables us to give examples of 3-component links with unknotted components and vanishing pairwise linking numbers, such that any two of these links have homology cobordant zero-surgeries in the above sense, but the zero-surgery manifolds are not homeomorphic. Moreover, the links are not concordant to one another, and in fact they can be chosen to be height $h$ but not height $h+1$ symmetric grope concordant, for each $h$ which is at least three.

##### Keywords
homology cobordism, zero-framed surgery, topological surgery, link concordance, symmetric grope concordance
##### Mathematical Subject Classification 2010
Primary: 57M25, 57N70