Chapter II History of the Smith Conjecture and Early Progress

Abstract

Publisher Summary This chapter discusses the history of the Smith conjecture and early progress. The study of periodic diffeomorphisms of the disk and the sphere began with the work of Brouwer [Bro] and Kerekjarto [Ke] in the 1920s. They proved that an orientation-preserving periodic diffeomorphism of the 2-disk or the 2-sphere was conjugate by a diffeomorphism to a rotation. These original proofs were incomplete and the gap was filled later by Eilenberg. Smith studied homeomorphisms of the n -disk and the n -sphere periodic of prime power period. He proved that the Z/ p -homology of the fixed point set is the same as that of a smaller dimensional disk or sphere. When the homeomorphism is orientation-preserving, then the codimension of the fixed point set is even. The analogue of the Smith conjecture in dimension m > 3 was investigated by Giffen, who showed that for each n > 1, m > 4 there exists a smooth m -sphere pair that is inequivalent to the standard pair.