Journal of the
Korean Mathematical Society JKMS

We study torsion phenomena in the integral cohomology of finite $H$-spaces $X$ through the Eilenberg--Moore spectral sequence converging to $H^*(\Omega X;Z_{p})$. We also investigate how the difference between the $Z_{p}$-filtration length $f_{p}(X)$ and the $Z_{p}$-cup length $c_{p}(X)$ on a simply connected finite $H$-space $X$ is reflected in the Eilenberg--Moore spectral sequence converging to $H^*(\Omega X;Z_{p})$. Finally we get the following result: Let $p$ be an odd prime and $X$ an $n$--connected finite $H$--space with dim $QH^{*}(X;Z_{p})$ $ \leq m$. Then $H^{*}(X;Z)$ is $p$--torsion free if $p\geq \frac{m-1}{n}$.

Keywords: finite $H$-space, cup length, filtration length, Eilenberg--Moore spectral sequence, $p$--torsion, loop space

MSC numbers: 55M30, 55P45, 55T20