Invariants and Structures of the Homology Cobordism Group of Homology Cylinders

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 ltrations by subgroups by developing new homomorphisms. First, we de ne extended Milnor invariants by combining the ideas of Milnor's link invariants and Johnson homomorphisms. They give rise to a descending ltration of the homology cobordism group of homology cylinders.We show that each successive quotient of the ltration is free abelian of nite rank. Second, we de ne 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 ltration is of in nite rank. Also we investigate further structures in the homology cobordism group of homology cylinders which previously known invariants do not detect.