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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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West’s problem on equivariant hyperspaces and Banach-Mazur compacta
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by Sergey Antonyan PDF
Trans. Amer. Math. Soc. 355 (2003), 3379-3404 Request permission


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) $\exp X$ is a $G$-ANR (resp., a $G$-AR); (c) $(\exp X)/G$ is an ANR (resp., an AR). This is applied to show that $(\exp G)/G$ is an ANR (resp., an AR) for each compact (resp., connected) Lie group $G$. If $G$ is a finite group, then $(\exp X)/G$ is a Hilbert cube whenever $X$ is a nondegenerate Peano continuum. Let $L(n)$ be the hyperspace of all centrally symmetric, compact, convex bodies $A\subset \mathbb {R}^n$, $n\ge 2$, 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 $H\subset O(n)$, $L_0(n)/H$ is a Hilbert cube manifold; (2) for each closed subgroup $K\subset O(n)$ 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 $(\exp S^{n-1})\setminus \{S^{n-1}\}$ have the same $O(n)$-homotopy type.
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Additional Information
  • Sergey Antonyan
  • Affiliation: Departamento de Matematicas, Facultad de Ciencias, UNAM, Ciudad Universitaria, México D.F. 04510, México
  • Email:
  • Received by editor(s): May 1, 2000
  • Received by editor(s) in revised form: September 15, 2002
  • Published electronically: April 8, 2003
  • Additional Notes: The author was supported in part by grant IN-105800 from PAPIIT (UNAM)
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 355 (2003), 3379-3404
  • MSC (2000): Primary 57N20, 57S10, 54B20, 54C55, 55P91, 46B99
  • DOI:
  • MathSciNet review: 1974693