A generalization of the Borsuk–Ulam theorem

Let $X$ be a connected paracompact Hausdorff space, acted on without fixed points by a cyclic group $\pi=\mathbf Z_p$ of prime order $p$. For any continuous mapping $f\colon X\to M$ let
$$\ A(f)=\{x\in X\mid f(x)=f(Tx)=\cdots=f(T^{p-1}x)\},$$
where $T$ is a generator of $\pi$.
Suppose $\Breve H^i(X;\mathbf Z_p)=0$ for $0<i<n$, and $M$ is a compact $\mathbf Z_p$-orientable topological manifold of dimension $m$. If the mapping $f^*\colon\Breve H^n(M;\mathbf Z_p)\to\Breve H^n(X;\mathbf Z_p)$ has zero image, then the cohomological dimension over $\mathbf Z_p$ of the set $A(f)$ is at least $n-m(p-1)$.
Furthermore, if $X$ is a generalized manifold of dimension $N$, and $n=m(p-1)$, then $\dim A(f)\geqslant N-m(p-1)$.
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