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 → 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

Primary: 57M25, 57M27

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