Homology cylinders of higher-order

TL;DR

This paper investigates the algebraic structures of submonoids and cobordism groups of homology cylinders over surfaces, revealing their complex relations and non-finite generation properties using Reidemeister torsion with non-commutative coefficients.

Contribution

It introduces new algebraic structures of homology cylinders related to higher solvable quotients and demonstrates their non-finite generation, expanding understanding beyond traditional mapping class groups.

Findings

Homology cobordism groups are larger than mapping class groups for surfaces with positive Betti number.

Submonoids of irreducible homology cylinders acting trivially on solvable quotients are not finitely generated.

Reidemeister torsion with non-commutative coefficients is used to analyze these structures.

Abstract

We study algebraic structures of certain submonoids of the monoid of homology cylinders over a surface and the homology cobordism groups, using Reidemeister torsion with non-commutative coefficients. The submonoids consist of ones whose natural inclusion maps from the boundary surfaces induce isomorphisms on higher solvable quotients of the fundamental groups. We show that for a surface whose first Betti number is positive, the homology cobordism groups are other enlargements of the mapping class group of the surface than that of ordinary homology cylinders. Furthermore we show that for a surface with boundary whose first Betti number is positive, the submonoids consisting of irreducible ones as 3-manifolds trivially acting on the solvable quotients of the surface group are not finitely generated.