The nontriviality of the first rational homology group of some connected invariant subsets of periodic transformations

Abstract

This note was inspired by some results of P. A. Smith [S]. One proves that for any periodic map of a manifold M M and any codimension two invariant submanifold P P of M M containing part of the stationary point set, connected invariant subsets of the complement of P P must carry nontrivial one-dimensional rational cycles, provided that M M satisfies some simple homological conditions (Theorem A). This fact has interesting consequences in transformation group theory.