Tree homology and a conjecture of Levine

TL;DR

This paper proves Levine's conjecture that a specific homomorphism between algebraic structures related to trees and quasi-Lie algebras is an isomorphism, advancing the understanding of topological invariants and link classifications.

Contribution

It confirms Levine's conjecture, establishing an isomorphism between T and D', and refines the relation between filtrations of homology cylinders.

Findings

Levine's conjecture is proven to be true.

The isomorphism between T and D' is established.

The relation between filtrations of homology cylinders is improved.

Abstract

In his study of the group of homology cylinders, J. Levine made the conjecture that a certain homomorphism eta': T -> D' is an isomorphism. Here T is an abelian group on labeled oriented trees, and D' is the kernel of a bracketing map on a quasi-Lie algebra. Both T and D' 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 the group of automorphisms of the free group. In this paper, we confirm Levine's conjecture. This is a central step in classifying the structure of links up to grope and Whitney tower concordance, as explained in other papers of this series. We also confirm and improve upon Levine's conjectured relation between two filtrations of the group of homology cylinders.