A Borsuk-Ulam theorem for $({\mathbb{Z}}_p{)}^k$-actions on products of (mod $p$) homology spheres

Abstract

It is proved that for a product action of $({\mathbb{Z}}_p{)}^k$ on a product of (mod p) homology spheres $N^{n_1}\times ...\times N^{n_k}$, where all $n_i$'s are assumed to be odd if $p$ is odd, and any continuous map $f:N^{n_1}\times ...\times N^{n_k}\to {\mathbb{R}}^m$ the set $A(f)=\{x\in N^{n_1}\times ...\times N^{n_k}|f(x)=f(gx)\forall g\in ({\mathbb{Z}}_p{)}^k\}$ has dimension at least $n_1+...+n_k-m(p^k-1)$, provided $n_i\ge m{p}^{i-1}(p-1)$ for all $i(1\le i\le k)$. Moreover, if $n_i\ge m{p}^{k-1}(p-1)$ for all $i(1\le i\le k)$ then the free action $\mu$ can be assumed arbitrary.

arXiv comment10 pages