• The Cohomology of Orbit Spaces of Certain Free Circle Group Actions

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/122/01/0079-0086

• # Keywords

Characteristic class; finitistic space; free action; index; spectral sequence.

• # 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 $\alpha^n\neq 0$, where $\alpha\in H^2(X/G;\mathbb{Z}_p)$ is the nonzero characteristic class of the $\mathbb{S}^1$-bundle $\mathbb{S}^1\hookrightarrow X\to X/G$. 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.

• # Author Affiliations

1. Department of Mathematics, University of Delhi, Delhi 110 007, India

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019