# Revista Matemática Iberoamericana

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**Volume 24, Issue 3, 2008, pp. 963–980**

**DOI: 10.4171/RMI/562**

Published online: 2008-12-31

Homology exponents for $H$-spaces

Alain Clément^{[1]}and Jérôme Scherer

^{[2]}(1) Vevey, Switzerland

(2) EPFL, Lausanne, Switzerland

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 $K(\mathbb Z,3)$.

*Keywords: *Homology exponent, *H*-space, loop space, Steenrod algebra

Clément Alain, Scherer Jérôme: Homology exponents for $H$-spaces. *Rev. Mat. Iberoamericana* 24 (2008), 963-980. doi: 10.4171/RMI/562