Homotopy-commutative 𝐻-spaces

Lin, James P.; Williams, Frank

doi:10.1090/S0002-9939-1991-1047005-2

Homotopy-commutative $H$-spaces
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by James P. Lin and Frank Williams PDF

Proc. Amer. Math. Soc. 113 (1991), 857-865 Request permission

Abstract:

Let $X$ be an $H$-space with ${H^*}(X;{Z_2}) \simeq {Z_2}[{x_1}, \ldots ,{x_d}] \otimes \Lambda ({y_1}, \ldots ,{y_d})$, where $\deg {x_i} = 4$ and ${y_i} = \operatorname {Sq}^1{x_i}$. In this article we prove that $X$ cannot be homotopy-commutative. Combining this result with a theorem of Michael Slack results in the following theorem: Let $X$ be a homotopy-commutative $H$-space with $\bmod 2$ cohomology finitely generated as an algebra. Then ${H^*}(X;{Z_2})$ is isomorphic as an algebra over $A(2)$ to the $\bmod 2$ cohomology of a torus producted with a finite number of $CP\left ( \infty \right )$s and $K({Z_{{2^{r,}}}}1)$s.

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A classification of homotopy commutative finitely generated

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