Two theorems from the theory of periodic transformations

Let $X$ be a connected paracompact Hausdorff space, freely acted on by a cyclic group of prime order $p$ with generator $T$. Let $f\colon X\to M$ be a continuous mapping of $X$ into a topological manifold $M$ of dimension $m$. Put $A(f)=\{x\in X\mid f(x)=f(Tx)=\dots=f(T^{p-1}x)\}$. If $M$ is orientable over $\mathbf Z_p$, $\check H^i(X;\mathbf Z_p)=0$ for $0<i<n$, and $f^*\colon\check H^m(M;\mathbf Z_p)\to\check H^m(X;\mathbf Z_p)$ has zero image, then, for $X$ weakly locally contractible, $\dim A(f)\geqslant n-m(p-1)$. If, in addition, $X$ is an $N$-dimensional topological manifold, then $\dim A(f)\geqslant N-m(p-1)$. For $p=2$, suppose $\check H^*(X;\mathbf Z_2)=H^*(S^n;\mathbf Z_2)$ and $\dim X\infty$, while $M$ is a connected compact closed manifold of dimension $n$ with a free involution $T'$. Let $A'(f)=\{x\in X \mid f(Tx)=T'f(x)\}$, and suppose $f^*\colon\check H^n(M;\mathbf Z_2)\to H^n(X;\mathbf Z_2)$ is a monomorphism. Then $A'(f)\ne\varnothing$.
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