The index of free circle actions in lens spaces (Q697598)

\textit{E. Fadell} and \textit{S. Husseini} [Ergodic Theory Dyn. Syst. 8, 73-85 (1988; Zbl 0657.55002)] defined an index for the action of a compact Lie group on a space. In the case of free \(S^1\)-actions, this index takes values in the integers, as follows. Let \(P_\infty\mathbb{C}\) denote the infinite complex projective space and \(c\in H^2(P_\infty\mathbb{C})\) the first universal Chern class. For a free \(S^1\)-action on a space \(X\), let \(c_X\) be the element of the cohomology of the orbit space which is the image of \(c\) under the homomorphism induced by the classifying map. Then the index of the action is the largest integer \(k\), if it exists, such that \((c_X)^k\neq 0\). For a natural number \(p\), let \(L^{2n+1}_p\) denote the lens space which is the orbit space of the free action of \(\mathbb{Z}/p\) on \(S^{2n+1}\) generated by the map \(z\to e^{2\pi i/p}z\). Denoting the orbit of \(z\) by \([z]\), there is a free action of \(S^1\) on \(L^{2n+1}_p\) defined by \(s\cdot[z]= [e^{2\pi ix/p}z]\) where \(e^{2\pi ix}= s\). The main result of the paper is that the Fadell-Husseini index of this action is the same as that of the standard action, defined by scalar multiplication, of \(S^1\) on \(S^{2n+1}\), namely, \(n\).