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

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.