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The Banach-Mazur Compactum Q(n) is an AR
by
Semeon Bogatyi
Moscow State University
Coauthors: Sergei Ageev (Moscow State University), Paul Fabel (Mississippi State University)
The first part of Problem 899 of J. West in ``Open problems in Topology'' is devoted to the following question: ``Is the set Q(n) of isometry classes of n-dimensional Banach spaces topologized by the metric d(E, F) = liminf{ ||T||·||T-1|| | T:E --> F is an isomorphism } an AR?''
Using conformal mappings the third author obtained a solution of this problem for n=2. The proof of inclusion Q(2) in AR was made in four steps, and it was possible to do the first step for general n. Only the following steps used the spaces of some conformal mappings. It is well known that Q(n) is a compactum and Q(n) ~ C(Rn)/GL(n), where C(Rn) is the hyperspace of all compact symmetric convex bodies in Rn with the Hausdorf metric and /GL(n) is the equivalence relation induced by the natural action of the group of all linear transformations GL(n). The first step of P. Fabel was in constracting of an embedding \sigma: C(Rn)/GL(n) --> C(Rn)/O(n) and of a retraction r : C(Rn)/O(n) --> C(Rn)/GL(n), where O(n) is the group of all orthogonal transformations. This step is crucial, as the question about the factorspace of an action of noncompact group was reduced to the question about the factorspace of an action of compact group. After Fabel's communication about the Banach-Mazur compactum Q(2) on Borsuk-Kuratowski Session in Varsaw on May 23 the first two authors understandied that the theory of equivariant retracts (with compact acting group) providies the tools for obtaining the solution of J. West problem in the case of general n. As for any bicompact group G the orbit space X/G is an AR-space for any metric G-AR space X, then the problem was reduced to prove that the some subspace of C(Rn) is an O(n)-AR space. The last statement can be proved by the consideration of the convex structure on the space C(Rn).
The authors feel deep indebtedness to T. Dobrowolski who encouraged the third author to study the Banach-Mazur compactum and to H. Toru\'nczyk for the invitation on the Borsuk-Kuratowski Session, what allowed the authors to meet and obtain the solution to the first part of J. West's problem in multidimensional case.
Theorem The Banach-Mazur compactum Q(n) is an AR.
Date received: June 24, 1996
Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-08.