#### 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