Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



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

Author: Sergey Antonyan
Journal: Trans. Amer. Math. Soc. 355 (2003), 3379-3404
MSC (2000): Primary 57N20, 57S10, 54B20, 54C55, 55P91, 46B99
Published electronically: April 8, 2003
Corrigendum: Trans. Amer. Math. Soc. 358 (2006), 5631-5633.
MathSciNet review: 1974693
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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.

References [Enhancements On Off] (What's this?)

  • 1. H. Abels, Parallelizability of proper actions, global $K$-slices and maximal compact subgroups, Math. Ann. 212 (1974), 1-19. MR 51:11460
  • 2. S. A. Antonyan, Retracts in categories of $G$-spaces, Izvestiya Akad. Nauk Arm. SSR. Ser. Matem. 15 (1980), 365-378; English transl. in: Soviet J. Contemp. Math. Anal. 15 (1980), 30-43. MR 82f:54027
  • 3. S. A. Antonyan, Equivariant generalization of Dugundji's theorem, Mat. Zametki 38 (1985), 608-616; English transl. in: Math. Notes 38 (1985), 844-848. MR 87a:54053
  • 4. S. A. Antonyan, An equivariant theory of retracts, in: Aspects of Topology (In Memory of Hugh Dowker), 251-269, London Math. Soc. Lecture Note Ser. 93, Cambridge Univ. Press, Cambridge, 1985. MR 87e:54090
  • 5. S. A. Antonian, Equivariant embeddings into $G$-AR's, Glasnik Matematicki 22 (42) (1987), 503-533. MR 89k:54041
  • 6. S. A. Antonyan, Retraction properties of an orbit space, Matem. Sbornik 137 (1988), 300-318; English transl. in: Math. USSR Sbornik 65 (1990), 305-321. MR 89k:54042
  • 7. S. A. Antonyan, Retraction properties of a space of orbits, II, Russian Math. Surv. 48 (1993), 156-157. MR 95b:54023
  • 8. S. A. Antonyan, The Banach-Mazur compacta are absolute retracts, Bull. Acad. Polon. Sci. Ser. Math. 46 (1998), 113-119. MR 99d:54020
  • 9. S. A. Antonyan, The topology of the Banach-Mazur compactum, Fund. Math. 166, no. 3 (2000), 209-232. MR 2001k:57026
  • 10. S. Banach, Théorie des Opérations Linéaires, Monografje Matematyczne, Warszawa, 1932. MR 97d:01035 (reprint)
  • 11. G. Bredon, Introduction to compact transformation groups, Academic Press, New York-London, 1972. MR 54:1265
  • 12. T. A. Chapman, Lectures on Hilbert cube manifolds, C. B. M. S. Regional Conference Series in Math., 28, Amer. Math. Soc., Providence, RI, 1975. MR 54:11336
  • 13. D. W. Curtis, Hyperspaces of noncompact metric spaces, Compositio Math. 40 (1980), 139-152. MR 81c:54009
  • 14. D. W. Curtis, Boundary sets in the Hilbert cube, Topol. Appl. 20 (1985), 201-221. MR 87d:57014
  • 15. J. Dugundji, Topology, Allyn and Bacon Inc., Boston, 1966. MR 33:1824
  • 16. P. Fabel, The Banach-Mazur compactum $Q(2)$ is an absolute retract, in: Topology and Applications (International Topological Conference dedicated to P. S. Alexandroff's 100th birthday, Moscow, May 27-31, 1996), p. 57, Moscow, 1996.
  • 17. R. E. Heisey and J. E. West, Orbit spaces of the hyperspace of a graph which are Hilbert cubes, Colloq. Math. 56 (1988), 59-69. MR 90a:57024
  • 18. D. W. Henderson, $Z$-sets in ANR's, Trans. Amer. Math. Soc. 213 (1975), 205-215. MR 52:11830
  • 19. A. Illanes and S. B. Nadler Jr., Hyperspaces. Fundamentals and Recent Advances, Marcel Dekker, Inc., New York-Basel, 1999. MR 99m:54006
  • 20. I. M. James and G. B. Segal On equivariant homotopy theory, Lecture Notes in Math. 788 (1980), 316-330. MR 82f:55014
  • 21. F. John. Extremum problems with inequalities as subsidiary conditions, in: F. John, Collected papers, 2 (ed. by J. Moser), 543-560, Birkhäuser, 1985. MR 10:719b; MR 87f:01107
  • 22. T. Matumoto, On $G$-$CW$ complexes and a theorem of J.H.C. Whitehead, J. Fac. Sci. Univ. Tokyo Sect. I A Math. 18 (1971), 363-374. MR 49:9842
  • 23. J. van Mill, Infinite-dimensional topology. Prerequisites and Introduction, North-Holland Publ. Co., Amsterdam-New York-Oxford-Tokyo, 1989. MR 90a:57025
  • 24. J. Milnor, Construction of universal bundles, II, Ann. Math. 63(3) (1956), 430-436. MR 17:1120a
  • 25. S. B. Nadler, Jr., Hyperspaces of sets, Marcel Dekker, Inc., New York and Basel, 1978. MR 58:18330
  • 26. R. Palais, The classification of $G$-spaces, Memoirs of the Amer. Math. Soc. 36, Providence, RI, 1960.
  • 27. E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966. MR 35:1007
  • 28. H. Torunczyk, On CE-images of the Hilbert cube and characterization of $Q$-manifolds, Fund. Math. 106 (1980), 31-40. MR 83g:57006
  • 29. H. Torunczyk and J. E. West, The fine structure of $S^1/S^1$; a $Q$-manifold hyperspace localization of the integers, in: Proc. Internat. Conf. Geom. Topol., 439-449, PWN-Pol. Sci. Publ., Warszawa, 1980. MR 83g:57005
  • 30. J. de Vries, Topics in the theory of topological transformation groups, in: Topological Structures II, pp. 291-304, Math. Centre Tracts, Vol. 116, Math. Centrum, Amsterdam, 1979. MR 81g:54045
  • 31. R. Webster Convexity, Oxford Univ. Press, Oxford, 1994. MR 98h:52001
  • 32. J. E. West, Infinite products which are Hilbert cubes, Trans. Amer. Math. Soc. 150 (1970), 1-25. MR 42:1055
  • 33. J. E. West, Induced involutions on Hilbert cube hyperspaces, Topology Proc. 1 (1976), 281-293. MR 58:24276
  • 34. J. E. West, Open problems in infinite-dimensional topology, in: Open Problems in Topology (ed. by J. van Mill and G. Reed), 524-586, North Holland, Amsterdam-New York-Oxford-Tokyo, 1990. MR 92c:54001
  • 35. M. Wojdyslawski, Rétractes absolus et hyperespaces des continus, Fund. Math. 32 (1939), 184-192.
  • 36. R. Y. T. Wong, Noncompact Hilbert cube manifolds, preprint.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57N20, 57S10, 54B20, 54C55, 55P91, 46B99

Retrieve articles in all journals with MSC (2000): 57N20, 57S10, 54B20, 54C55, 55P91, 46B99

Additional Information

Sergey Antonyan
Affiliation: Departamento de Matematicas, Facultad de Ciencias, UNAM, Ciudad Universitaria, México D.F. 04510, México

Keywords: Banach-Mazur compacta, $G$-ANR, $Q$-manifold, hyperspace, orbit space, homotopy type, $G$-nerve
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)
Article copyright: © Copyright 2003 American Mathematical Society