  We say that a space X admits a homology exponent if there exists an exponent
for the torsion subgroup of the integral homology. Our main result states if an
H-space of finite type admits a homology exponent, then either it is, up to
2-completion, a product of spaces of the form BZ/2^r, S^1, K(Z, 2), and K(Z,3),
or it has infinitely many non-trivial homotopy groups and k-invariants. We then
show with the same methods that simply connected $H$-spaces whose mod 2
cohomology is finitely generated as an algebra over the Steenrod algebra do not
have homology exponents, except products of mod 2 finite H-spaces with copies
of K(Z, 2) and K(Z,3).
