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
is an isomorphism.
Both
and
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 . 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
and
range
of Levine’s map.
The isomorphism
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.