Abstract
The aim of this paper is to affirmatively resolve a long-standing conjecture
in algebraic topology, related to the free p-rank of abelian groups acting
freely on products of spheres. To be precise, let (Z=p)^r act freely on
S^n1 ×S^n2 ×· · ·×S^nl for a prime p and let l_o denote the number of odd dimensional
spheres in the product of spheres. In this paper we show that,
if p is odd prime, then r is less than or equal to l_o and the free p-rank
of S^n1 × S^n2 × · · · × S^nl is equal to l_o. On the other hand, if p is equal
to 2, it is shown that r is less than or equal to l and the free p-rank of
S^n1 ×S^n2 ×· · ·×S^nl is equal to l.