Characterization of Y₂-Equivalence for Homology Cylinders

Abstract

For Σ a compact connected oriented surface, we consider homology cylinders over Σ: these are homology cobordisms with an extra homological triviality condition. When considered up to Y 2 -equivalence, which is a surgery equivalence relation arising from the Goussarov-Habiro theory, homology cylinders form an Abelian group. In this paper, when Σ has one or zero boundary component, we define a surgery map from a certain space of graphs to this group. This map is shown to be an isomorphism, with inverse given by some extensions of the first Johnson homomorphism and Birman-Craggs homomorphisms.