Filtering smooth concordance classes of topologically slice knots

Cochran, Tim D; Harvey, Shelly; Horn, Peter

doi:10.2140/gt.2013.17.2103

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

Filtering smooth concordance classes of topologically slice knots

Tim D Cochran, Shelly Harvey and Peter Horn

Geometry & Topology 17 (2013) 2103–2162

DOI: 10.2140/gt.2013.17.2103

Abstract

We propose and analyze a structure with which to organize the difference between a knot in $S^{3}$ bounding a topologically embedded $2$ –disk in $B^{4}$ and it bounding a smoothly embedded disk. The $n$ –solvable filtration of the topological knot concordance group, due to Cochran–Orr–Teichner, may be complete in the sense that any knot in the intersection of its terms may well be topologically slice. However, the natural extension of this filtration to what is called the $n$ –solvable filtration of the smooth knot concordance group, is unsatisfactory because any topologically slice knot lies in every term of the filtration. To ameliorate this we investigate a new filtration, ${ℬ_{n}}$ , that is simultaneously a refinement of the $n$ –solvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. We show that each $ℬ_{n} ∕ ℬ_{n + 1}$ has infinite rank. But our primary interest is in the induced filtration, ${T_{n}}$ , on the subgroup, $T$ , of knots that are topologically slice. We prove that $T ∕ T_{0}$ is large, detected by gauge-theoretic invariants and the $τ$ , $s$ , $ϵ$ –invariants, while the nontriviality of $T_{0} ∕ T_{1}$ can be detected by certain $d$ –invariants. All of these concordance obstructions vanish for knots in $T_{1}$ . Nonetheless, going beyond this, our main result is that $T_{1} ∕ T_{2}$ has positive rank. Moreover under a “weak homotopy-ribbon” condition, we show that each $T_{n} ∕ T_{n + 1}$ has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity.

Keywords

concordance, slice knot, 4–manifold

Mathematical Subject Classification 2010

Primary: 57M25

References

Publication

Received: 1 June 2012

Accepted: 1 April 2013

Published: 19 July 2013

Proposed: Robion Kirby

Seconded: Walter Neumann, Ronald Fintushel

Authors

Tim D Cochran
Department of Mathematics Rice University MS 136, PO Box 1892 Houston, TX 77251-1892 USA
http://math.rice.edu/~cochran
Shelly Harvey
Department of Mathematics Rice University MS 136, PO Box 1892 Houston, TX 77251-1892 USA
http://math.rice.edu/~shelly
Peter Horn
Department of Mathematics Syracuse University 215 Carnegie Building Syracuse, NY 13244-1150 USA
https://pdhorn.expressions.syr.edu/

Volume 17, issue 4 (2013)