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Covering link calculus and the bipolar filtration of topologically slice links

Jae Choon Cha and Mark Powell

Geometry & Topology 18 (2014) 1539–1579

The bipolar filtration introduced by T Cochran, S Harvey and P Horn is a framework for the study of smooth concordance of topologically slice knots and links. It is known that there are topologically slice 1–bipolar knots which are not 2–bipolar. For knots, this is the highest known level at which the filtration does not stabilize. For the case of links with two or more components, we prove that the filtration does not stabilize at any level: for any n, there are topologically slice links which are n–bipolar but not (n + 1)–bipolar. In the proof we describe an explicit geometric construction which raises the bipolar height of certain links exactly by one. We show this using the covering link calculus. Furthermore we discover that the bipolar filtration of the group of topologically slice string links modulo smooth concordance has a rich algebraic structure.

covering link calculus, concordance, bipolar filtration
Mathematical Subject Classification 2010
Primary: 57M25, 57N70
Received: 1 May 2013
Accepted: 7 October 2013
Published: 7 July 2014
Proposed: Peter Teichner
Seconded: Robion Kirby, Yasha Eliashberg
Jae Choon Cha
Department of Mathematics
Pohang University of Science and Technology
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
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