A Borsuk-Ulam theorem for (Zp)k-actions on products of (mod p) homology spheres

Let p be a prime number. It is proven that for the product of free actions of Z(p) on mod p homology spheres N-ni, i = 1,..., k, where n(i) s are assumed to be odd if p is odd, for any continuous map f : N-n1 x... x N-nk --> R-m the set A (f) = {x is an element of N-n1 x... x N-nk/f(x) = f (gx) for all g is an element of (Z(p))(k)) has dimension at least n(1) +...+ n(k) - m(p(k) - 1), provided n(i) >= mp(i-1)(p - 1) for all i (1 <= i <= k). (C) 2006 Elsevier B.V. All rights reserved.