Homology exponents for $H$-spaces

We say that a space X admits a homology exponent if there exists an exponentfor the torsion subgroup of the integral homology. Our main result states if anH-space of finite type admits a homology exponent, then either it is, up to2-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 thenshow with the same methods that simply connected $H$-spaces whose mod 2cohomology is finitely generated as an algebra over the Steenrod algebra do nothave homology exponents, except products of mod 2 finite H-spaces with copiesof K(Z, 2) and K(Z,3).