Abstract
We say that a space
X
admits a \emph{homology exponent} if there exists an exponent for the torsion subgroup of
H^*(X;\mathbb Z)
. Our main result states that 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
B\mathbb Z/2^r
,
S^1
,
\mathbb C P^\infty
, and
K(\mathbb Z,3)
, or it has infinitely many non-trivial homotopy groups and
k
-invariants. Relying on recent advances in the theory of
H
-spaces, we then show 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
\mathbb C P^\infty
and
K(\mathbb Z,3)
.