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
Milestones
Received: 28 October 2013
Revised: 6 December 2013
Accepted: 16 December 2013
Published: 9 October 2014
Authors
Jae Choon Cha
Department of Mathematics
Pohang University of Science and Technology
Gyungbuk
Pohang 790–784
South Korea
School of Mathematics
Korea Institute for Advanced Study
Seoul 130–722
South Korea
Mark Powell
Department of Mathematics
Indiana University
Rawles Hall
831 East 3rd Street
Bloomington, IN 47405
United States