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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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$\phi$-Postnikov systems and extensions of $H$-spaces
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by Albert O. Shar
Proc. Amer. Math. Soc. 49 (1975), 237-244
DOI: https://doi.org/10.1090/S0002-9939-1975-0367994-X

Abstract:

Let $f:X \to Y$ be a map of CW complexes and let $\pi :{F_f} \to X$ be the fibration induced by $f$. The following theorems are proven: Theorem. Assume $F( = {F_f})$ and $\Omega Y$ are simply connected and that (a) ${f^ \ast }:{H^n}(Y;{\pi _n}(Y)) \to {H^n}(X;{\pi _n}(Y))$ is epic for all $n$, (b) ${(i \wedge i)^ \ast }:{H^n}(F \wedge F;\operatorname {co} \ker {f_{{n^ \ast }}}) \to {H^n}(\Omega Y \wedge \Omega Y;\operatorname {co} \ker {f_{{n^ \ast }}})$ is monic for all $n$ (where ${f_{{n^ \ast }}}:{\pi _n}(X) \to {\pi _n}(Y))$). If $X$ is an $H$-space then $F$ is an $H$-space such that $\pi :F \to X$ is an $H$-map. Theorem. Assume $Y$ is $(p - 1)$-connected, $F$ is $(q - 1)$-connected $(p - 1 \geq 2,q - 1 \geq 1)$ of dimension $< \min (2p - 1,p + q - 1)$, and ${H^ \ast }(Y)$ is free. If $X$ is an $H$-space and ${f^ \ast }:{H^n}(Y) \to {H^n}(X)$ is onto for all $n$ then $F$ is an $H$-space and the map $\pi :F \to X$ is an $H$-map. Analogous theorems are shown to hold for loop spaces.
References
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Bibliographic Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 49 (1975), 237-244
  • MSC: Primary 55D45
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0367994-X
  • MathSciNet review: 0367994