Nonconcordant links with homology cobordant zero-framed surgery manifolds

Jae Choon Cha and Mark Powell

Vol. 272 (2014), No. 1, 1–33

DOI: 10.2140/pjm.2014.272.1

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