The topology of the Banach–Mazur compactum

Sergey Antonyan

Fundamenta Mathematicae (2000)

  • Volume: 166, Issue: 3, page 209-232
  • ISSN: 0016-2736

Abstract

top
Let J(n) be the hyperspace of all centrally symmetric compact convex bodies A ⊆ ℝ n , n ≥ 2, for which the ordinary Euclidean unit ball is the ellipsoid of maximal volume contained in A (the John ellipsoid). Let J 0 ( n ) be the complement of the unique O(n)-fixed point in J(n). We prove that: (1) the Banach-Mazur compactum BM(n) is homeomorphic to the orbit space J(n)/O(n) of the natural action of the orthogonal group O(n) on J(n); (2) J(n) is an O(n)-AR; (3) J 0 ( 2 ) / S O ( 2 ) is an Eilenberg-MacLane space 𝐊 ( ℚ , 2 ) ; (4) B M 0 ( 2 ) = J 0 ( 2 ) / O ( 2 ) is noncontractible; (5) BM(2) is a nonhomogeneous absolute retract. Other models for BM(n) are established.

How to cite

top

Antonyan, Sergey. "The topology of the Banach–Mazur compactum." Fundamenta Mathematicae 166.3 (2000): 209-232. <http://eudml.org/doc/212478>.

@article{Antonyan2000,
abstract = {Let J(n) be the hyperspace of all centrally symmetric compact convex bodies $A ⊆ \mathbb \{R\}^n$, n ≥ 2, for which the ordinary Euclidean unit ball is the ellipsoid of maximal volume contained in A (the John ellipsoid). Let $J_0(n)$ be the complement of the unique O(n)-fixed point in J(n). We prove that: (1) the Banach-Mazur compactum BM(n) is homeomorphic to the orbit space J(n)/O(n) of the natural action of the orthogonal group O(n) on J(n); (2) J(n) is an O(n)-AR; (3) $J_0(2)/SO(2)$ is an Eilenberg-MacLane space $\mathbf \{K\}(\mathbb \{Q\},2)$; (4) $BM_0(2) = J_0(2)/O(2)$ is noncontractible; (5) BM(2) is a nonhomogeneous absolute retract. Other models for BM(n) are established.},
author = {Antonyan, Sergey},
journal = {Fundamenta Mathematicae},
keywords = {Banach-Mazur compactum; G-ANR; orbit space; Q-manifoldhomotopy type; Eilenberg-MacLane space $\mathbf \{K\}(\mathbb \{Q\},2)$; Hilbert cube},
language = {eng},
number = {3},
pages = {209-232},
title = {The topology of the Banach–Mazur compactum},
url = {http://eudml.org/doc/212478},
volume = {166},
year = {2000},
}

TY - JOUR
AU - Antonyan, Sergey
TI - The topology of the Banach–Mazur compactum
JO - Fundamenta Mathematicae
PY - 2000
VL - 166
IS - 3
SP - 209
EP - 232
AB - Let J(n) be the hyperspace of all centrally symmetric compact convex bodies $A ⊆ \mathbb {R}^n$, n ≥ 2, for which the ordinary Euclidean unit ball is the ellipsoid of maximal volume contained in A (the John ellipsoid). Let $J_0(n)$ be the complement of the unique O(n)-fixed point in J(n). We prove that: (1) the Banach-Mazur compactum BM(n) is homeomorphic to the orbit space J(n)/O(n) of the natural action of the orthogonal group O(n) on J(n); (2) J(n) is an O(n)-AR; (3) $J_0(2)/SO(2)$ is an Eilenberg-MacLane space $\mathbf {K}(\mathbb {Q},2)$; (4) $BM_0(2) = J_0(2)/O(2)$ is noncontractible; (5) BM(2) is a nonhomogeneous absolute retract. Other models for BM(n) are established.
LA - eng
KW - Banach-Mazur compactum; G-ANR; orbit space; Q-manifoldhomotopy type; Eilenberg-MacLane space $\mathbf {K}(\mathbb {Q},2)$; Hilbert cube
UR - http://eudml.org/doc/212478
ER -

References

top
  1. [1] H. Abels, Parallelizability of proper actions, global K-slices and maximal compact subgroups, Math. Ann. 212 (1974), 1-19. Zbl0276.57019
  2. [2] S. A. Antonyan, Retracts in categories of G-spaces, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 15 (1980), 365-378 (in Russian); English transl.: Soviet J. Contemp. Math. Anal. 15 (1980), 30-43. Zbl0468.57026
  3. [3] S. A. Antonyan, An equivariant theory of retracts, in: Aspects of Topology (In Memory of Hugh Dowker), London Math. Soc. Lecture Note Ser. 93, Cambridge Univ. Press, 1985, 251-269. 
  4. [4] S. A. Antonyan, Retraction properties of the orbit space, Mat. Sb. 137 (1988), 300-318 (in Russian); English transl.: Math. USSR-Sb. 65 (1990), 305-321. 
  5. [5] S. A. Antonyan, Equivariant embeddings and free G-spaces, in: Simpozionul VI Tiraspolean Topol. Gener. Aplic., Institutul Pedagogic din Tiraspol, Chişinău, 1991, 9-10. 
  6. [6] S. A. Antonyan, The Banach-Mazur compacta are absolute retracts, Bull. Polish Acad. Sci. Math. 46 (1998), 113-119. Zbl0906.54012
  7. [7] G. Bredon, Introduction to Compact Transformation Groups, Academic Press, 1972. Zbl0246.57017
  8. [8] G. Bredon, Sheaf Theory, 2nd ed., Springer, 1997. 
  9. [9] T. A. Chapman, Lectures on Hilbert Cube Manifolds, CBMS Regional Conf. Ser. in Math. 28, Amer. Math. Soc., 1976. 
  10. [10] A. Dvoretsky, Some results on convex bodies and Banach spaces, in: Proc. Internat. Sympos. on Linear Spaces (Hebrew Univ. of Jerusalem), 1960, 123-160. 
  11. [11] R. E. Edwards, Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, 1965. Zbl0182.16101
  12. [12] P. Fabel, The Banach-Mazur compactum Q(2) is an absolute retract, talk at the Moscow Internat. Topology Conf. in memory of P. S. Alexandrov, May 27-31, 1996. 
  13. [13] D. W. Henderson, Z-sets in ANR's, Trans. Amer. Math. Soc. 213 (1975), 205-215. Zbl0315.57004
  14. [14] G. Hochshild, The Structure of Lie Groups, Holden-Day, 1965. 
  15. [15] D. Husemoller, Fiber Bundles, McGraw-Hill, 1966. 
  16. [16] I. M. James and G. B. Segal, On equivariant homotopy theory, in: Lecture Notes Math. 788, Springer, 1980, 316-330. Zbl0439.57021
  17. [17] J. Jaworowski, Extension properties of G-maps, in: Proc. Internat. Conf. Geometric Topology (Warszawa, 1978), K. Borsuk and A. Kirkor (eds.), PWN, Warszawa, 1980, 209-213. Zbl1144.55005
  18. [18] F. John, Extremum problems with inequalities as subsidiary conditions, in: F. John, Collected Papers, Vol. 2 (ed. by J. Moser), Birkhäuser, 1985, 543-560. 
  19. [19] J. Lindenstrauss and V. D. Milman, The local theory of normed spaces and its applications to convexity, in: Handbook of Convex Geometry, P. M. Gruber and J. M. Wills (eds.), North-Holland, Amsterdam, 1993, 1149-1220. Zbl0791.52003
  20. [20] R. Palais, The classification of G-spaces, Mem. Amer. Math. Soc. 36 (1960). Zbl0119.38403
  21. [21] R. Palais, On the existence of slices for actions of noncompact Lie groups, Ann. of Math. 73 (1961), 295-323. Zbl0103.01802
  22. [22] M. Steinberger and J. West, On the geometric topology of locally linear actions of finite groups, in: Geometric and Algebraic Topology, Banach Center Publ. 18, PWN, Warszawa, 1986, 181-204. Zbl0652.57024
  23. [23] D. Sullivan, Geometric Topology, Part I, Massachusetts Institute of Technology, 1970 (mimeographed notes). 
  24. [24] D. Sullivan, De Rham homotopy theory, mimeographed lecture notes of a course given in Orsay University in 1973/1974, Orsay, France, 1974. 
  25. [25] H. Toruńczyk, On CE-images of Hilbert cube and characterization of Q-manifolds, Fund. Math. 106 (1980), 31-40. 
  26. [26] H. Toruńczyk and J. E. West, Fine structure of S 1 / S 1 ; a Q-manifold hyperspace localization of integers, in: Proc. Internat. Conf. Geometric Topology, PWN, Warszawa, 1980, 439-449. Zbl0469.57012
  27. [27] J. de Vries, Topics in the theory of topological transformation groups, in: Topological Structures II, Math. Centre Tracts 116, Math. Centrum, Amsterdam, 1979, 291-304. 
  28. [28] R. Webster, Convexity, Oxford Univ. Press, 1994. 
  29. [29] J. E. West, Infinite products which are Hilbert cubes, Trans. Amer. Math. Soc. 150 (1970), 1-25. Zbl0198.56001
  30. [30] J. E. West, Open problems of infinite-dimensional topology, in: Open Problems in Topology, J. van Mill and G. Reed (eds.), North-Holland, 1990, 524-586. 
  31. [31] R. Y. T. Wong, Noncompact Hilbert cube manifolds, preprint. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.