Abstract:
This paper considers generalizations of the Bourgin–Yang theorem. It is shown that if $f\colon X\to M$ is a continuous mapping of a paracompact free
$\mathbf Z_p$-space $X$ into an $m$-dimensional manifold $M$, then, under the condition that $\operatorname{in}X\geqslant n>m(p-1)$ (where $\operatorname{in}X$ is the index in the sense of Yang) and $f^*V_i=0$ for $i\geqslant1$, where the $V_i$ are the Wu classes of $M$, the following inequality holds:
$$
\operatorname{in}\{x\in X\mid f(x)=f(gx)\ \forall g\in\mathbf Z_p\}\geqslant n-m(p-1).
$$
Besides this result, certain “nonsymmetric” versions of the Borsuk–Ulam theorem are proved.
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