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Homotopy-commutative $ H$-spaces


Authors: James P. Lin and Frank Williams
Journal: Proc. Amer. Math. Soc. 113 (1991), 857-865
MSC: Primary 55P45; Secondary 55S05, 55S45
DOI: https://doi.org/10.1090/S0002-9939-1991-1047005-2
MathSciNet review: 1047005
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1047005-2
Keywords: Homotopy-commutative $ H$-space, cohomology operation, Steenrod algebra
Article copyright: © Copyright 1991 American Mathematical Society

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