Chapter X Finite Group Actions on Homotopy 3-Spheres

Abstract

The chapter discusses the finite group actions on homotopy 3-spheres. The Smith conjecture states that certain types of smooth actions of a cyclic group on a homotopy 3-sphere are essentially linear. This hypothesis is proved in a substantial number of cases based on theorems presented in the chapter. It is an immediate consequence of the Schönflies theorem and the definition of essentially linear that any essentially linear action on S 3 is equivariantly diffeomorphic to a linear action. The chapter introduces the notion of an orbifold coined by Thurston to describe spaces that are locally the quotients of finite group actions. The chapter discusses two- and three-dimensional examples of these spaces. The language of orbifolds is a convenient framework for dealing with the quotient spaces of properly discontinuous group actions. The connection between finite group actions and linear actions through the intermediary of Seifert-fibered orbifolds is established.