Construction of semi-free S3 actions on homotopy sphere with untwisted fixed point set

TL;DR

This paper extends surgery techniques from S1 to S3 actions on homotopy spheres, aiming to construct semi-free actions with untwisted fixed point sets, and discusses related open problems.

Contribution

It develops a theory for semi-free S3 actions on homotopy spheres, generalizing Browder's work on S1 actions and exploring new construction methods.

Findings

Extended surgery techniques to S3 actions

Constructed semi-free actions with untwisted fixed points

Raised open problems in the theory of group actions

Abstract

William Browder in his paper "Surgery and the theory of differentiable transformation groups" developed surgery techniques to study semi-free actions of S1 on homotopy spheres, under the additional assumption that the fixed point set is a homotopy sphere. He used this surgery to show how to construct such actions. In this paper, I discussed a similar theory for semi-free actions of S3 on homotopy spheres. An open problem is raised at the end of the paper.