#### Volume 16, issue 1 (2012)

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Tree homology and a conjecture of Levine

### James Conant, Rob Schneiderman and Peter Teichner

Geometry & Topology 16 (2012) 555–600
##### 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 ${\eta }^{\prime }:\mathsc{T}\to {D}^{\prime }$ is an isomorphism. Both $\mathsc{T}$ and ${D}^{\prime }$ 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 $Out\left({F}_{n}\right)$. 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 $\mathsc{T}$ and range ${D}^{\prime }$ of Levine’s map.

The isomorphism ${\eta }^{\prime }$ 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
##### Publication
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/