In 1955, Klee proved that every convex body in infinite-dimensional Hilbert
space has periodic homeomorphisms of all periods with fixed point sets homeomorphic
to arbitrary separable complete metric spaces. I shall discuss joint work with J. van Mill
in the context of arbitrary compact convex infinite-dimensional subsets C of Fre'chet
spaces. We have these theorems:
Theorem 1: (A partial converse to a theorem of P. A. Smith) If p is prime, then every
compact metric Absolute Neighborhood Retract (ANR) with trivial Z/pZ homology is
(homeomorphic to) the fixed point set of a periodic homeomorphism of C with period p.
(Essentially presented to the seminar last year.)
Let G be a compact metric group.
Theorem 2: G acts semifreely on C with arbitrary fixed point and uniformly small orbits.
Theorem 3: All such G-actions on C are (equivariantly) isotopic.
Theorem 4: If X is a compact metric space with the shape (pro-homotopy type) of a
point, then there is a semifree G-action on C with fixed point set homeomorphic to X.