Sergey A. Antonyan

Extensorial properties of orbit spaces of proper group actions

Abstract Extensorial properties of orbit spaces of locally compact proper group actions are investigated.

West's problem on equivariant hyperspaces and Banach-Mazur compacta

Let G be a compact Lie group, X a metric G-space, and exp X the hyperspace of all nonempty compact subsets of X endowed with the Hausdorff metric topology and with the induced action of G. We prove that the following three assertions are equivalent: (a) X is locally continuum-connected (resp., connected and locally continuum-connected); (b) expX is a G-ANR (resp., a G-AR); (c) (exp X)/G is an ANR (resp., an AR). This is applied to show that (expG)/G is an ANR (resp., an AR) for each compact (resp., connected) Lie group G. If G is a finite group, then (expX)/G is a Hilbert cube whenever X is a nondegenerate Peano continuum. Let L(n) be the hyperspace of all centrally symmetric, compact, convex bodies A C R n , n > 2, for which the ordinary Euclidean unit ball is the ellipsoid of minimal volume containing A, and let L 0 (n) be the complement of the unique O(n)-fixed point in L(n). We prove that: (1) for each closed subgroup H C O(n), L 0 (n)/H is a Hilbert cube manifold; (2) for each closed subgroup K C O(n) acting non-transitively on S n-1 , the K-orbit space L(n)/K and the K-fixed point set L(n)[K] are Hilbert cubes. As an application we establish new topological models for tha Banach-Mazur compacta L(n)/O(n) and prove that L 0 ( n ) and (exp S n-1 ) \ {S n-1 } have the same O(n)-homotopy type.

Invariant pseudometrics on Palais proper G-spaces

Let G be a locally compact Hausdorff group. It is proved that: 1. on each Palais proper G-space X there exists a compatible family of G-invariant pseudometrics; 2.the existence of a compatible G-invariant metric on a metrizable proper G-space X is equivalent to the paracompactness of the orbit space X/G; 3. if in addition G is either almost connected or separable, and X is locally separable, then there exists a compatible G-invariant metric on X.

A short proof that hyperspaces of Peano continua are absolute retracts

We give a short proof of Wojdyslawski’s famous theorem. Theorem (Wojdyslawski [6]). Let X be a Peano continuum. Then the hyperspace 2 of all nonempty compact subsets of X is an absolute retract for metric spaces. This result is an essential step in the proof of the Curtis-Schori-West Hyperspace Theorem to the effect that 2 is a Hilbert cube for any Peano continuum X (see, e.g., the book of van Mill [5, §8.4]). Wojdyslawski’s original proof is rather complicated [6]. A simpler proof was suggested later on by Kelley [4], which is, however, based on a difficult Lefschetz-Dugundji characterization of metric ANR’s (see [5, Theorem 5.2.1]). Yet another proof, also based on the Lefschetz-Dugundji characterization, can be found in [5, §5.3]. Our proof is elementary and it does not rely on the Lefschetz-Dugundji criterion. Proof. Let d be any compatible metric on X and let dH be the Hausdorff metric on 2 . Assume that (Y, ρ) is a metric space, A is a closed subset of Y and f : A→ 2 is a continuous map. Following [3], choose a canonical cover ω of Y \A in Y , that is to say: (1) ω is an open cover of Y \ A, locally finite in Y \ A; (2) for each neighborhood V of a point a ∈ A in Y there exists a neighborhood S of a in Y contained in V , such that every element U ∈ ω which meets S is contained in V . We note that the second condition implies that every neighborhood of any boundary point of A in Y contains infinitely many open sets in ω (see [2, Ch. III, §1]). Let N (ω) denote the nerve of ω endowed with the CW topology. We will denote by pU the vertex of N (ω) corresponding to U ∈ ω. Then according to [3], there exist a Hausdorff space Z and a continuous map μ : Y → Z with the following properties: (a) Z as a set coincides with the disjoint union A ∪ N (ω); (b) A is closed in Z and the restriction μ|A is the identical homeomorphism; (c) Z \A=N (ω) is taken with its CW topology and μ(Y \A) ⊂ Z \ μ(A); (d) a base of neighborhoods of a ∈ A in Z is determined by selecting a neighborhood W of a in Y and taking in Z the set W ∩ A together with the closed Received by the editors October 29, 1999 and, in revised form, September 5, 2000. 2000 Mathematics Subject Classification. Primary 54B20; Secondary 54C55.

Bourgin-Yang-type theorem for

A variant of the Bourgin-Yang theorem for a-compact perturbations of a closed linear operator (in general, unbounded and having an infinite-dimensional kernel) is proved. An application to integrodifferential equations is discussed.

a

In what follows, G is a Hausdorff topological group with identity element e. A topological transformation group or a G-space with phase group G is a triple \(\langle X, G, \alpha \rangle \), where X is a topological space and \(\alpha \).