The topology of the Banach–Mazur compactum

• Autorzy: Sergey A. Antonyan
• Języki publikacji: EN

Abstrakty

EN

Let J(n) be the hyperspace of all centrally symmetric compact convex bodies $A ⊆ \Bbb R^n$, 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 $\bold K(\Bbb Q,2)$; (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.

Słowa kluczowe

EN

Banach-Mazur compactum; G-ANR; orbit space; Q-manifoldhomotopy type; Eilenberg-MacLane space $\bold K(\Bbb Q,2)$

Kategorie tematyczne

• 46B99: None of the above, but in this section
• 57S10: Compact groups of homeomorphisms
• 55P20: Eilenberg-Mac Lane spaces
• 55P92: Relations between equivariant and nonequivariant homotopy theory
• 57N20: Topology of infinite-dimensional manifolds
• 54C55: Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2000
• Tom: 166
• Numer: 3
• Strony: 209-232

Daty

wydano

2000

otrzymano

1997-09-23

poprawiono

1998-11-17

poprawiono

2000-06-12

Twórcy

autor

Sergey A. Antonyan

• Departamento de Matemáticas, Facultad de Ciencias, UNAM, Circuito Exterior, C.U., México, D.F. 04510, México

Bibliografia

• [1] H. Abels, Parallelizability of proper actions, global K-slices and maximal compact subgroups, Math. Ann. 212 (1974), 1-19.
• [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.
• [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] 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] 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] S. A. Antonyan, The Banach-Mazur compacta are absolute retracts, Bull. Polish Acad. Sci. Math. 46 (1998), 113-119.
• [7] G. Bredon, Introduction to Compact Transformation Groups, Academic Press, 1972.
• [8] G. Bredon, Sheaf Theory, 2nd ed., Springer, 1997.
• [9] T. A. Chapman, Lectures on Hilbert Cube Manifolds, CBMS Regional Conf. Ser. in Math. 28, Amer. Math. Soc., 1976.
• [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] R. E. Edwards, Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, 1965.
• [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] D. W. Henderson, Z-sets in ANR's, Trans. Amer. Math. Soc. 213 (1975), 205-215.
• [14] G. Hochshild, The Structure of Lie Groups, Holden-Day, 1965.
• [15] D. Husemoller, Fiber Bundles, McGraw-Hill, 1966.
• [16] I. M. James and G. B. Segal, On equivariant homotopy theory, in: Lecture Notes Math. 788, Springer, 1980, 316-330.
• [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.
• [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] 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.
• [20] R. Palais, The classification of G-spaces, Mem. Amer. Math. Soc. 36 (1960).
• [21] R. Palais, On the existence of slices for actions of noncompact Lie groups, Ann. of Math. 73 (1961), 295-323.
• [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.
• [23] D. Sullivan, Geometric Topology, Part I, Massachusetts Institute of Technology, 1970 (mimeographed notes).
• [24] D. Sullivan, De Rham homotopy theory, mimeographed lecture notes of a course given in Orsay University in 1973/1974, Orsay, France, 1974.
• [25] H. Toruńczyk, On CE-images of Hilbert cube and characterization of Q-manifolds, Fund. Math. 106 (1980), 31-40.
• [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.
• [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] R. Webster, Convexity, Oxford Univ. Press, 1994.
• [29] J. E. West, Infinite products which are Hilbert cubes, Trans. Amer. Math. Soc. 150 (1970), 1-25.
• [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] R. Y. T. Wong, Noncompact Hilbert cube manifolds, preprint.