The cohomology of orbit spaces of certain free circle group actions

Suppose that (G =mathbb{S}^1) acts freely on a finitistic space X whose (mod p) cohomology ring is isomorphic to that of a lens space (L^{2m-1}(p;q_1,ldots,q_m)) or (mathbb{S}^1times mathbb{C}P^{m-1}). The mod p index of the action is defined to be the largest integer n such that α ⁿ ?≠?0, where (alpha ,epsilon, H^2(X/G;mathbb{Z}_p)) is the nonzero characteristic class of the (mathbb{S}^1)-bundle (mathbb{S}^1hookrightarrow Xrightarrow X/G). We show that the mod p index of a free action of G on (mathbb{S}^1times mathbb{C}P^{m-1}) is p???1, when it is defined. Using this, we obtain a Borsuk–Ulam type theorem for a free G-action on (mathbb{S}^1times mathbb{C}P^{m-1}). It is note worthy that the mod p index for free G-actions on the cohomology lens space is not defined.