Let G = S^d, d = 0, 1 or 3, act freely on a finitistic connected space X.
This paper gives the cohomology classification of X if a mod 2 or rational
cohomology of the orbit space X/G is isomorphic to the product of a projective
space and sphere FP^n x S^m, where F = R, C or H, respectively. For a free
involution on X, a lower bound of covering dimension of the coincidence set of
a continuous map f: X -> R^k is also determined