Fixed-point sets of transformation groups on infinite-product spaces.

Abstract

then a is said to be a semifree action of G on X with fixed-point set Y. If Y is empty, then a is said to be free. The problem addressed in this paper is to find sufficient conditions that each closed subset of X be the fixed-point set of a semifree action of G. The main result proven here is Theorem 3.2: If X is a separable, infinite-dimensional Fr&chet space and G is a topological group for which there is an effective action on X as a transformation group, then each closed subset of X is the fixedpoint set of a semifree action of G on X. This generalizes the theorem of V. L. Klee, Jr. [8] that each compactum of a separable, infinitedimensional Hilbert space is the fixed-point sef of periodic homeomorphisms of the space onto itself of all periods greater than one, and is complementary to A. Beck's theorem [4] that if X is a metric space on which there is an action of the real line without degenerate orbits, then each closed set of X is the union of the degenerate orbits of some action of the real line on X (in the sense that whereas Beck's theorem removes restrictions on the space X, the theorem of this paper removes restrictions on the group G).