Homology exponents for $H$-spaces

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 $K(\mathbb Z,3)$.