The cohomology of orbit spaces of certain free circle group actions

Abstract

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 $\mathbb{S}^1\times \mathbb{C}P^{m-1}$. 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 $\mathbb{S}^1\hookrightarrow X\rightarrow X/G$. We show that the mod p index of a free action of G on $\mathbb{S}^1\times \mathbb{C}P^{m-1}$ is p − 1, when it is defined. Using this, we obtain a Borsuk–Ulam type theorem for a free G-action on $\mathbb{S}^1\times \mathbb{C}P^{m-1}$. It is note worthy that the mod p index for free G-actions on the cohomology lens space is not defined.