Fixed point Free Actions of Spheres and Equivariant maps

Abstract

This paper generalizes the concept of index and co-index and some related results for free actions of G = S 0 on a paracompact Hausdorff space which were introduced by Conner and Floyd[1]. We define the index and co-index of a finitistic free G-space X, where G = S d , 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 S 2n+1 (resp. (4n+3)-sphere S 4n+3 ) for the action of componentwise multiplication of G = S 1 (resp. S 3 ) is n.We also determine the orbit spaces of free actions of G = S 3 on a finitistic space X with the mod 2 cohomology and the rational cohomology product of spheres S n × S m , 1 ≤ n ≤ m. 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.