## Geometry & Topology

### Tree homology and a conjecture of Levine

#### 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 $η′:T→D′$ 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 $Out(Fn)$. 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.

#### Article information

Source
Geom. Topol., Volume 16, Number 1 (2012), 555-600.

Dates
Revised: 16 January 2012
Accepted: 16 January 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732389

Digital Object Identifier
doi:10.2140/gt.2012.16.555

Mathematical Reviews number (MathSciNet)
MR2916294

Zentralblatt MATH identifier
1284.57007

#### Citation

Conant, James; Schneiderman, Rob; Teichner, Peter. Tree homology and a conjecture of Levine. Geom. Topol. 16 (2012), no. 1, 555--600. doi:10.2140/gt.2012.16.555. https://projecteuclid.org/euclid.gt/1513732389

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