The cobordism group of homology cylinders

TL;DR

This paper studies the algebraic structure of the homology cobordism group of homology cylinders over a surface, revealing it has an infinitely generated abelianization, thus not being perfect, especially when the surface has positive Betti number or multiple boundary components.

Contribution

It demonstrates that the abelianization of the homology cobordism group is infinitely generated and has infinite rank in certain cases, answering longstanding questions.

Findings

The abelianization is infinitely generated for surfaces with positive Betti number.

The abelianization has infinite rank when the surface has multiple boundary components.

Results apply to the homology cylinder analogue of the Torelli group.

Abstract

Garoufalidis and Levine introduced the homology cobordism group of homology cylinders over a surface. This group can be regarded as a generalization of the mapping class group. Using torsion invariants, we show that the abelianization of this group is infinitely generated provided that the first Betti number of the surface is positive. In particular, this shows that the group is not perfect. This answers questions of Garoufalidis-Levine and Goda-Sakasai. Furthermore we show that the abelianization of the group has infinite rank for the case that the surface has more than one boundary component. These results hold for the homology cylinder analogue of the Torelli group as well.