Geometry & Topology

Tree homology and a conjecture of Levine

James Conant, Rob Schneiderman, and Peter Teichner

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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:TD 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

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

Received: 13 December 2010
Revised: 16 January 2012
Accepted: 16 January 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57N10: Topology of general 3-manifolds [See also 57Mxx]

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


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.

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