Tree homology and a conjecture of Levine

Conant, James; Schneiderman, Rob; Teichner, Peter

doi:10.2140/gt.2012.16.555

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Tree homology and a conjecture of Levine

James Conant, Rob Schneiderman and Peter Teichner

Geometry & Topology 16 (2012) 555–600

DOI: 10.2140/gt.2012.16.555

Abstract

In his study of the group of homology cylinders, J Levine [Algebr. Geom. Topol. 2 (2002) 1197–1204] made the conjecture that a certain group homomorphism $η^{'} : T \to D^{'}$ is an isomorphism. Both $T$ and $D^{'}$ are defined combinatorially using trivalent trees and have strong connections to a variety of topological settings, including the mapping class group, homology cylinders, finite type invariants, Whitney tower intersection theory and the homology of $Out (F_{n})$ . In this paper, we confirm Levine’s conjecture by applying discrete Morse theory to certain subcomplexes of a Kontsevich-type graph complex. These are chain complexes generated by trees, and we identify particular homology groups of them with the domain $T$ and range $D^{'}$ of Levine’s map.

The isomorphism $η^{'}$ is a key to classifying the structure of links up to grope and Whitney tower concordance, as explained in [Proc. Natl. Acad. Sci. USA 108 (2011) 8131–8138; arXiv 1202.3463]. In this paper and [arXiv 1202.2482] we apply our result to confirm and improve upon Levine’s conjectured relation between two filtrations of the group of homology cylinders.

Keywords

Levine conjecture, tree homology, homology cylinder, Whitney tower, discrete Morse theory, quasi-Lie algebra

Mathematical Subject Classification 2010

Primary: 57M27, 57M25

Secondary: 57N10

References

Publication

Received: 13 December 2010

Revised: 16 January 2012

Accepted: 16 January 2012

Published: 8 April 2012

Proposed: Rob Kirby

Seconded: David Gabai, Cameron Gordon

Authors

James Conant
Department of Mathematics University of Tennessee Knoxville TN 37996 USA

Rob Schneiderman
Department of Mathematics and Computer Science Lehman College, CUNY 250 Bedford Park Boulevard West Bronx NY 10468-1589 USA
http://comet.lehman.cuny.edu/schneiderman/
Peter Teichner
Department of Mathematics University of California, Berkeley and MPIM Bonn Berkeley CA 94720-3840 USA
http://math.berkeley.edu/~teichner/

Volume 16, issue 1 (2012)