A Borsuk–Ulam theorem for -actions on products of (modp) homology spheres

Let p be a prime number. It is proven that for the product of free actions of Z p on modp homology spheres N n i , i = 1 , … , k , where n i s are assumed to be odd if p is odd, for any continuous map f : N n 1 × ⋯ × N n k → R m the set A ( f ) = { x ∈ N n 1 × ⋯ × N n k | f ( x ) = f ( g x ) ∀ g ∈ ( Z p ) k } has dimension at least n 1 + ⋯ + n k − m ( p k − 1 ) , provided n i ⩾ m p i − 1 ( p − 1 ) for all i ( 1 ⩽ i ⩽ k ).