Cohomology algebra of the orbit space of free circle group actions on lens spaces

TL;DR

This paper computes the mod p cohomology ring of the orbit space resulting from a free circle group action on lens spaces, revealing the structure and index related to the characteristic class.

Contribution

It determines the mod p cohomology ring of the orbit space and establishes the mod p index for free circle actions on lens spaces, extending understanding of their topological properties.

Findings

Cohomology ring of the orbit space is explicitly computed.

Mod p index of the action is p-1 when the characteristic class is nonzero.

Provides conditions under which the index is defined and its value.

Abstract

Suppose that G=S^1 acts freely on a finitistic space X whose mod p cohomology ring isomorphic to that of a lens space L^{2m-1}(p;q_1,...,q_m). In this paper, we determine the mod p cohomology ring of the orbit space X/G. If the characteristic class \alpha\belongs H^2(X/G;Z_p) of the S^1-bundle S--> X--> X/G is nonzero, then mod p ndex of the action is deined to be the largest integer n such that \alpha^n is nonzero. We also show that the mod p index of a free action of S^1 on a lens space L^(2m-1)(p;q_1,...,q_m) is p-1, provided that \alpha is nonzero.