Fixed point Free Actions of Spheres and Equivariant maps

TL;DR

This paper extends the concepts of index and co-index for free sphere actions, providing new bounds and classifications for orbit spaces, and deriving Borsuk-Ulam type results for these actions.

Contribution

It generalizes index theory for free actions of spheres S^1 and S^3, and determines orbit spaces and bounds for these actions on various cohomological spaces.

Findings

Index of X is at most the mod 2 cohomology index.

Orbit spaces of free S^3 actions on cohomology spheres are characterized.

Upper bounds for the index lead to Borsuk-Ulam type theorems.

Abstract

This paper generalizes the concept of index and co-index and some related results for free actions of G = S0 on a paracompact Hausdorff space which were introduced by Conner and Floyd. We define the index and co-index of a finitistic free G-space X, where G = Sd , d = 1 or 3 and prove that the index of X is not more than the mod 2 cohomology index of X. We observe that the index and co-index of a (2n + 1)-sphere (resp. (4n+3)-sphere) for the action of componentwise multiplication of G = S1 (resp. S3) is n. We also determine the orbit spaces of free actions of G = S3 on a finitistic space X with the mod 2 cohomology and the rational cohomology product of spheres. The orbit spaces of circle actions on the mod 2 cohomology X is also discussed. Using these calculation, we obtain an upper bound of the index of X and the Borsuk-Ulam type results.