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 ${\eta }^{\prime }:\mathcal{T}\to D^{\prime }$ is an isomorphism. Both $\mathcal{T}$ 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 $\mathcal{T}$ 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.