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Absorbing sets in infinite-dimensional manifolds. (English) Zbl 1147.54322
Mathematical Studies Monograph Series 1. L’viv: VNTL Publishers (ISBN 5-7773-0061-8). 240 p. (1996).
Publisher’s description: The method of absorbing sets goes back to Anderson, Bessaga and Pełczyński, Toruńczyk, West. In its modern form, due to Bestvina and Mogilski, it is one of the most powerful tools in infinite-dimensional topology.
The book gives a self-contained exposition of basic and advanced theory of absorbing spaces, new constructions of absorbing sets for various classes of spaces defined by means of dimensional and descriptive set-theoretical conditions, and a wide spectrum of applications to function spaces, topological groups, hyperspaces, convex sets, spaces of probability measures, linear topological spaces.
Contents: Chapter I. Basic Theory
1.1. Preliminaries. 1.2. Homotopy dense and homotopy negligible sets. 1.3. The strong discrete approximation property. 1.4. \(Z\)-Sets and strong \(Z\)-Sets. 1.5. Strong universality. 1.6. Absorbing and coabsorbing spaces. 1.7. Strongly universal and absorbing pairs.
Chapter II. Constructions of absorbing spaces
2.1. Preliminaries I: Descriptive Set Theory. 2.2. Preliminaries II: Dimension Theory. 2.3. Invertible and soft maps. 2.4. Absorbing spaces for \([0, 1]\)-stable classes. 2.5. Weak inverse limits and absorbing sets. 2.6. Some negative results.
Chapter III. Advanced Topics
3.1. Interplay between strongly universal spaces and pairs. 3.2 Characterizing (strong) \(C\)-universality for “nice” classes \(C\).
Chapter IV. Applications I.
4.1. Infinite products. 4.2. Topological groups. 4.3. Hyperspaces.
Chapter V. Applications II: Convex Sets.
5.1. Locally compact convex sets. 5.2. Topologically complete convex sets. 5.3. Strong universality in convex sets. 5.4. Strong universality in locally convex spaces. 5.5. Some counterexamples. 5.6. Spaces of probability measures. 5.7. Function spaces \(C_p(X)\) and \(C_p^*(X)\).

54H20 Topological dynamics (MSC2010)
54-02 Research exposition (monographs, survey articles) pertaining to general topology
57Nxx Topological manifolds
58Bxx Infinite-dimensional manifolds