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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the realization of invariant subgroups of $ \pi \sb\ast (X)$


Author: A. Zabrodsky
Journal: Trans. Amer. Math. Soc. 285 (1984), 467-496
MSC: Primary 55Q52; Secondary 55P45, 55S45
MathSciNet review: 752487
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Abstract: Let $ p$ be a prime and $ T:X \to X$ a self map. Let $ A$ be a multiplicatively closed subset of the algebraic closure of $ {F_p}$. Denote by $ {V_{T,A}}$ the set of characteristic values of $ {\pi_{\ast} }(T) \otimes {F_p}$ lying in $ A$. It is proved that under certain conditions $ {V_{T,A}}$ is realizable by a pair $ \tilde X,\tilde T$: There exist a space $ \tilde X$, maps $ \tilde T:\tilde X \to \tilde X$ and $ f:\tilde X\: \to \:X$ so that $ f\,\circ \,\tilde T\sim T\,\circ \,f,{\pi _ * }(F)$ is $ \bmod\, p$ injective and $ {\rm {im}}({\pi_{\ast} }(f) \otimes {F_p}) = {V_{T,A}}$. This theorem yields, among others, examples of spaces whose $ \bmod\, p$ cohomology rings are polynomial algebras.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0752487-8
PII: S 0002-9947(1984)0752487-8
Article copyright: © Copyright 1984 American Mathematical Society