The Cohomology of Orbit Spaces of Certain Free Circle Group Actions

Proceedings Mathematical Sciences

Hemant Kumar Singh²  Tej Bahadur Singh¹ 

[1] $$;Department of Mathematics, University of Delhi, Delhi 0 00, India$$

关键词: Characteristic class;    finitistic space;    free action;    index;    spectral sequence.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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 $mathbb{S}^1×mathbb{C}P^{m-1}$. The mod 𝑝 index of the action is defined to be the largest integer 𝑛 such that $𝛼^n≠ 0$, where $𝛼in H^2(X/G;mathbb{Z}_p)$ is the nonzero characteristic class of the $mathbb{S}^1$-bundle $mathbb{S}^1hookrightarrow X→ X/G$. We show that the mod 𝑝 index of a free action of 𝐺 on $mathbb{S}^1×mathbb{C}P^{m-1}$ is 𝑝-1, when it is defined. Using this, we obtain a Borsuk–Ulam type theorem for a free 𝐺-action on $mathbb{S}^1×mathbb{C}P^{m-1}$. It is note worthy that the mod 𝑝 index for free 𝐺-actions on the cohomology lens space is not defined.

【 授权许可】

Unknown