Cohomology Classification of Spaces with Free S3 Actions

TL;DR

This paper classifies the cohomology types of spaces with free S3 actions, showing they are either high-dimensional spheres or products involving quaternion projective spaces, extending similar results for S1 actions.

Contribution

It provides a cohomology classification for spaces with free S3 actions and characterizes their topological structure, including cases with S1 actions.

Findings

Spaces with free S3 actions are either (4n+3)-spheres or products of 3-spheres and quaternion projective spaces.

The orbit space's cohomology is that of quaternionic projective space HPn.

Similar classifications are discussed for S1 actions.

Abstract

This paper gives the cohomology classification of finitistic spaces X equipped with free actions of the group G = S3 and the orbit space X/G is the integral or mod 2 cohomology quaternion projective space HPn. We have proved that X is the integral or mod 2 cohomology (4n+3)-sphere or the product of 3-sphere and quaternion projective space HPn. Similar results for G = S1 actions are also discussed.