Mappings of free $\mathbf Z_p$-spaces into manifolds

A. Yu. Volovikov

Izvestiya. Mathematics , Tome 20 (1983) no. 1, pp. 35-53

Résumé

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. Bibliography: 16 titles.