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Let G be a finite group acting freely on a finitistic space X having cohomology type (0, b) (for example, $\mathbb S$n × $\mathbb S$2n is a space of type (0, 1) and the one-point union $\mathbb S$n ∨ $\mathbb S$2n ∨ $\mathbb S$3n is a space of type (0, 0)). It is known that a finite group G that contains ℤp ⊕ ℤp ⊕ ℤp, p a prime, cannot act freely on $\mathbb S$n × $\mathbb S$2n. In this paper, we show that if a finite group G acts freely on a space of type (0, 1), where n is odd, then G cannot contain ℤp ⊕ ℤp, p an odd prime. For spaces of cohomology type (0, 0), we show that every p-subgroup of G is either cyclic or a generalized quaternion group. Moreover, for n even, it is shown that ℤ2 is the only group that can act freely on X.

The classical Borsuk Ulam theorem can be stated as: there exists no equivariant map S n → S n − 1 , relative to the antipodal actions on the spheres. Let G = Z 2 act freely on a finitistic space X with mod 2 cohomology ring isomorphic to that of the product of a projective space (real, complex or quaternionic) and the 3-sphere. In this paper, we show that the Volovikov's index of R P m × S 3 is any one of the integers 2, 4, m + 3 or m + 4 . In case of C P m × S 3 , this index is 3, 4 or 2 m + 4 and that of H P m × S 3 is 4, 5, 8 or 9. We apply this to determine the possibilities of nonexistence of equivariant maps X → S n or S n → X .

Let $$G=\mathbb {Z}_p,$$ $$p>2$$ a prime, act freely on a finitistic space X with mod p cohomology ring isomorphic to that of $$\mathbb {F}P^m\times \mathbb {S}^3$$ , where $$m+1\not \equiv 0$$ mod p and $$\mathbb {F}=\mathbb {C}$$ or $$\mathbb {H}$$ . We wish to discuss the nonexistence of G-equivariant maps $$\mathbb {S}^{2q-1}\rightarrow X$$ and $$ X\rightarrow \mathbb {S}^{2q-1}$$ , where $$\mathbb {S}^{2q-1}$$ is equipped with a free G-action. These results are analogues of the celebrated Borsuk-Ulam theorem. To establish these results first we find the cohomology algebra of orbit spaces of free G-actions on X. For a continuous map $$f\!:\! X\rightarrow \mathbb {R}^n$$ , a lower bound of the cohomological dimension of the partial coincidence set of f is determined. Furthermore, we approximate the size of the zero set of a fibre preserving G-equivariant map between a fibre bundle with fibre X and a vector bundle. An estimate of the size of the G-coincidence set of a fibre preserving map is also obtained. These results are parametrized versions of the Borsuk-Ulam theorem.