Tree homology and a conjecture of Levine
James Conant, Rob Schneiderman, Peter Teichner · Geometry & Topology · 2012
In his study of the group of homology cylinders, J Levine [23] made the conjecture that a certain group homomorphism Á 0 W T !D 0 is an isomorphism.Both T and D 0 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 0 of Levine's map.The isomorphism Á 0 is a key to classifying the structure of links up to grope and Whitney tower concordance, as explained in [6; 5].In this paper and [3] we apply our result to confirm and improve upon Levine's conjectured relation between two filtrations of the group of homology cylinders.