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Absolute extensors and functors in the asymptotic category
by
Oksana Shukel'
Lviv National University, 1 Universytetska Str., 79000 Lviv, Ukraine
The asymptotic category A is defined as follows [1]. The objects of A are proper metric spaces and the morphims are the proper asymptotically Lipschitz maps. Recall that a map f:(X, d)→(Y, r) of metric spaces is called asymptotically Lipschitz if there exist l > 0, s ≥ 0 such that r(f(x), f(y)) ≤ ld(x, y)+s for every x, y ∈ X.
The Kuratowski symbol X tY can be naturally defined as follows. Given two proper metric spaces, X and Y, the symbol XtY means that every proper asymptotically Lipschitz map f:A→ Y defined on a closed subset A of X admits an extension over X which is also an asymptotically Lipschitz map.
We say that a proper metric space X is an absolute extensors for a class C of proper metric spaces of YtX, for every Y ∈ C. If C coincides with the class of all proper metric spaces (of the asymptotic dimension ≤ n, we say that X is an absolute extensor (in asymptotic dimension n), see [3]. The n-th hypersymmetric power functor expn and the G-symmetric power functor SPGn are examples of normal functors in the asymptotic category A, see [2]. The main results of the talk concern preservation of the classes of absolute extensors (in asymptotic dimension n) by the mentioned G-symmetric power functors and the n-th hypersymmetric power functors.
These results are counterparts of the corresponding results of Basmanov on preservation of absolute extensors by covariant functors of finite degree in the category of compact Hausdorf spaces [4].
References:
[1] Dranishnikov A. Asymptotic topology // Russian Math. Surveys. 55:6 (2000), 71-116.
[2] Shukel O. Functors of finite degree and asymptotic dimension zero. // Matem. Studii. 29:1 (2008), 101-107.
[3] Gromov M. Asymptotic invariants for infinite groups. LMS Lecture Notes. 182:2 (1993).
[4] Basmanov V. N. Covariant functors, retracts, and dimension. // Dokl. Akad. Nauk SSSR 271:5 (1983), 1033-1036. (in Russian).
Date received: May 13, 2008
Copyright © 2008 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 # cawm-65.