Fixed point Free Actions of Spheres and Equivariant maps

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.