  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.
