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
–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.
