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

Author:

Antonyan Sergey

Abstract

Let G G be a compact Lie group, X X a metric G G -space, and exp X \exp X the hyperspace of all nonempty compact subsets of X X endowed with the Hausdorff metric topology and with the induced action of G G . We prove that the following three assertions are equivalent: (a) X X is locally continuum-connected (resp., connected and locally continuum-connected); (b) exp X \exp X is a G G -ANR (resp., a G G -AR); (c) ( exp X ) / G (\exp X)/G is an ANR (resp., an AR). This is applied to show that ( exp G ) / G (\exp G)/G is an ANR (resp., an AR) for each compact (resp., connected) Lie group G G . If G G is a finite group, then ( exp X ) / G (\exp X)/G is a Hilbert cube whenever X X is a nondegenerate Peano continuum. Let L ( n ) L(n) be the hyperspace of all centrally symmetric, compact, convex bodies A R n A\subset \mathbb {R}^n , n 2 n\ge 2 , for which the ordinary Euclidean unit ball is the ellipsoid of minimal volume containing A A , and let L 0 ( n ) L_0(n) be the complement of the unique O ( n ) O(n) -fixed point in L ( n ) L(n) . We prove that: (1) for each closed subgroup H O ( n ) H\subset O(n) , L 0 ( n ) / H L_0(n)/H is a Hilbert cube manifold; (2) for each closed subgroup K O ( n ) K\subset O(n) acting non-transitively on S n 1 S^{n-1} , the K K -orbit space L ( n ) / K L(n)/K and the K K -fixed point set L ( n ) [ K ] L(n)[K] are Hilbert cubes. As an application we establish new topological models for tha Banach-Mazur compacta L ( n ) / O ( n ) L(n)/O(n) and prove that L 0 ( n ) L_0(n) and ( exp S n 1 ) { S n 1 } (\exp S^{n-1})\setminus \{S^{n-1}\} have the same O ( n ) O(n) -homotopy type.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference36 articles.

1. Parallelizability of proper actions, global 𝐾-slices and maximal compact subgroups;Abels, Herbert;Math. Ann.,1974

2. Retracts in categories of 𝐺-spaces;Antonjan, S. A.;Izv. Akad. Nauk Armyan. SSR Ser. Mat.,1980

3. Equivariant generalization of Dugundji’s theorem;Antonyan, S. A.;Mat. Zametki,1985

4. An equivariant theory of retracts;Antonian, Sergey,1985

5. Equivariant embeddings into 𝐺-ARs;Antonian, S.;Glas. Mat. Ser. III,1987

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