• Fulltext

       

        Click here to view fulltext PDF


      Permanent link:
      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

       

      Hemant Kumar Singh1 Tej Bahadur Singh1

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

       
  • Proceedings – Mathematical Sciences | News

    • Editorial Note on Continuous Article Publication

      Posted on July 25, 2019

      Click here for Editorial Note on CAP Mode

© 2021-2022 Indian Academy of Sciences, Bengaluru.