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The homotopy groups of spaces whose cohomology is a $ Z\sb{p}$ truncated polynomial algebra


Author: Albert Shar
Journal: Proc. Amer. Math. Soc. 38 (1973), 172-178
MSC: Primary 55E05
DOI: https://doi.org/10.1090/S0002-9939-1973-0310877-X
MathSciNet review: 0310877
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Abstract: On such a space one can define a ``Hopf'' invariant homomorphism $ h:{\pi _{qn - 1}}(K) \to Z$ in two ways. We prove both definitions are equivalent and show that $ {}_p{\pi _i}(K) \simeq {}_p{\pi _{i - 1}}({S^{n - 1}}) \oplus {}_p{\pi _i}({S^{qn - 1}})$ if and only if there is an $ \alpha \in {\pi _{qn - 1}}(K)$ such that $ (h(\alpha ),p) = 1$. As immediate corollaries of this we get a result of Toda on the homotopy groups of the reduced product spaces of spheres and a well-known result of Serre on the odd primary parts of the homotopy groups of spheres.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0310877-X
Keywords: Hopf invariant, Whitehead product, Serre spectral sequence, reduced product space
Article copyright: © Copyright 1973 American Mathematical Society

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