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 $M$ and any codimension two invariant submanifold $P$ of $M$ containing part of the stationary point set, connected invariant subsets of the complement of $P$ must carry nontrivial one-dimensional rational cycles, provided that $M$ satisfies some simple homological conditions (Theorem A). This fact has interesting consequences in transformation group theory.