Homology cobordism group of homology cylinders and invariants related to lower central series

TL;DR

This paper explores the structure of the homology cobordism group of homology cylinders, extending classical invariants and establishing a filtration related to free nilpotent groups, revealing new relations and invariants.

Contribution

It introduces extended Johnson, Milnor, and Orr invariants for homology cylinders and analyzes their relations and the associated filtration's structure.

Findings

Established a combined filtration via kernels of extended invariants.

Determined the images of invariants under the filtration.

Counted the number of linearly independent invariants.

Abstract

The homology cobordism group of homology cylinders is a generalization of both the mapping class group of surfaces and the string link concordance group. We consider extensions of Johnson homomorphisms of a mapping class group, Milnor invariants and Orr invariants of links to homology cylinders, all of which are related to free nilpotent groups. We establish a combined filtration via kernels of extended Johnson homomorphisms and extended Milnor invariants. We determine its image under the three invariants, and investigate relations among the invariants, and relations of the filtration to automorphism groups of free nilpotent groups and to graded free Lie algebras. We obtain the number of linearly independent invariants by examining the successive quotients of the filtration.