Characterization of Y_2-equivalence for homology cylinders

TL;DR

This paper characterizes the structure of homology cylinders over surfaces with one or zero boundary components under Y_2-equivalence, establishing an isomorphism with a graph-based space and connecting it to Johnson and Birman-Craggs homomorphisms.

Contribution

It introduces a surgery map from a graph space to the homology cylinder group and proves it is an isomorphism, extending known homomorphisms.

Findings

The surgery map is an isomorphism for surfaces with ≤1 boundary component.

Homology cylinders form an Abelian group under Y_2-equivalence.

Extensions of Johnson and Birman-Craggs homomorphisms characterize the inverse map.

Abstract

For S a compact connected oriented surface, we consider homology cylinders over S: 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 S 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.