Geodesic
A generalization of the Borsuk–Ulam theorem
Sbornik. Mathematics, Tome 36 (1980) no. 2, pp. 195-202 
A. Yu. Volovikov. A generalization of the Borsuk–Ulam theorem. Sbornik. Mathematics, Tome 36 (1980) no. 2, pp. 195-202. http://geodesic.mathdoc.fr/item/SM_1980_36_2_a3/
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     title = {A~generalization of the {Borsuk{\textendash}Ulam} theorem},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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, 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 $\dim A(f)\geqslant N-m(p-1)$. Bibliography: 8 titles.

[1] D. G. Bourgin, Modern algebraic topology, MacMillan, New York, 1963 | MR

[2] P. Godeman, Algebraicheskaya topologiya i teoriya puchkov, IL, Moskva, 1961

[3] P. Konner, E. Floid, Gladkie periodicheskie otobrazheniya, izd-vo “Mir”, Moskva, 1969

[4] H. J. Munkholm, “Borsuk–Ulam type theorems for proper $\mathbf Z_p$-actions on (mod $p$ homology) $n$-spheres”, Math. Scand., 24:2 (1969), 167–185 | MR | Zbl

[5] M. Nakaoka, “Generalizations of Borsuk–Ulam theorem”, Osaka J. Math., 7:2 (1970), 423–441 | MR | Zbl

[6] G. S. Skordev, “On a theorem of Bourgin–Yang”, SERDICA, Bulg. math, publ., 1 (1975), 183–198 | MR | Zbl

[7] G. E. Bredon, Sheaf theory, McGraw-Hill, New York, 1967 | MR | Zbl

[8] E. G. Sklyarenko, “K teorii obobschennykh mnogoobrazii”, Izv. AN SSSR, seriya matem., 35:4 (1971), 831–843 | Zbl