Invariants and structures of the homology cobordism group of homology cylinders

TL;DR

This paper introduces new invariants and filtrations for the homology cobordism group of homology cylinders, revealing complex algebraic structures and infinite rank abelianizations not detectable by previous invariants.

Contribution

It develops extended Milnor invariants and Hirzebruch-type invariants, providing new tools to analyze the structure of the homology cobordism group of homology cylinders.

Findings

Each filtration quotient is free abelian of finite rank

The abelianization of the intersection of filtrations has infinite rank

New invariants detect previously unseen structures

Abstract

The homology cobordism group of homology cylinders is a generalization of the mapping class group and the string link concordance group. We study this group and its filtrations by subgroups by developing new homomorphisms. First, we define extended Milnor invariants by combining the ideas of Milnor's link invariants and Johnson homomorphisms. They give rise to a descending filtration of the homology cobordism group of homology cylinders. We show that each successive quotient of the filtration is free abelian of finite rank. Second, we define Hirzebruch-type intersection form defect invariants obtained from iterated p-covers for homology cylinders. Using them, we show that the abelianization of the intersection of our filtration is of infinite rank. Also we investigate further structures in the homology cobordism group of homology cylinders which previously known invariants do not detect.