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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Primitive obstructions in the cohomology of loopspaces


Author: Frank Williams
Journal: Proc. Amer. Math. Soc. 91 (1984), 477-480
MSC: Primary 55P35; Secondary 55P45, 55S20
MathSciNet review: 744652
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Abstract: Let $ X$ and $ X'$ be $ H$-spaces. If $ f:\Omega X \to \Omega X'$ is an $ H$-map then the obstruction to $ f$ being a homotopy-commutative map is a subset $ \left\{ {{c_2}(f)} \right\} \subset \left[ {\Omega X\Lambda \Omega X;{\Omega ^2}X'} \right]$. In this paper we prove: $ If[f]$ is in the image of the composition

$\displaystyle \left[ {{P_{k + m}}\Omega X;X'} \right] \to \left[ {\Sigma \Omega X;X'} \right]\mathop \to \limits^ \approx \left[ {\Omega X;\Omega X'} \right],$

then $ \left\{ {{c_2}(f)} \right\}$ is in the image of the composition

$\displaystyle \left[ {{P_k}\Omega X\Lambda {P_m}\Omega X;X'} \right] \to \left[... ...p \to \limits^ \approx \left[ {\Omega X\Lambda \Omega X;{\Omega ^2}X'} \right].$

Consequently if $ \alpha \in {H^n}(\Omega X;{Z_p})$ is an $ {A_3}$-class in the sense of Stasheff then each element of $ \left\{ {{c_2}(f)} \right\}$ is of the form $ \sum {{{c'}_i}} \otimes {c''_i}$ where the $ {c''_i}$ are primitive.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0744652-6
PII: S 0002-9939(1984)0744652-6
Keywords: $ H$-space, homotopy-commutativity, obstruction
Article copyright: © Copyright 1984 American Mathematical Society