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

 

$ \phi$-Postnikov systems and extensions of $ H$-spaces


Author: Albert O. Shar
Journal: Proc. Amer. Math. Soc. 49 (1975), 237-244
MSC: Primary 55D45
MathSciNet review: 0367994
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0367994-X
PII: S 0002-9939(1975)0367994-X
Keywords: Postnikov system, $ H$-space, $ H$-map, obstruction
Article copyright: © Copyright 1975 American Mathematical Society