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 $H^{\ast }(X;\mathbb{Z})$. 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 $K(\mathbb{Z},3)$.