Chapter I The Smith Conjecture

Abstract

This chapter discusses the Smith conjecture. The proof of the Smith conjecture represents a culmination of the efforts of many mathematicians. The work of Smith on cyclic group actions was seminal. The chapter also describes the formulations and generalizations of the Smith conjecture. The arguments proving the Smith conjecture can easily be adapted for the piecewise linear (PL) case or for the topological case, provided that in the topological case it is assumed that the fixed point set is locally flat. The techniques used to establish the Smith conjecture can be used to prove various generalizations. The theorem on the solution of the Smith conjecture affirms several special cases of the Poincaré conjecture. The applications of these ideas and methods to three-dimensional topology are due to Thurston, Meeks, and Yau, with help from Bass, Shalen, Gordon, and Litherland.