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Inverse spectra. (English) Zbl 0934.54001

North-Holland Mathematical Library. 53. Amsterdam: Elsevier. x, 421 p. (1996).
This book significantly extends and updates the previously published Russian book [V. V. Fedorchuk and A. I. Chigogidze, Absolute retracts and infinite-dimensional manifolds (1992; Zbl 0762.54017)]; responding to a suggestion to translate it, the author instead decided to rewrite the book completely in order to include several new topics and recent developments.
The work of R. Haydon [Studia Math. 52, 23-31 (1974; Zbl 0294.46016)] was a landmark in the development of the method of inverse spectra. Haydon proved that the class of compact absolute extensors in dimension 0 coincides with the class of Dugundji compacta. But undoubtedly the most significant ingredient of his work was the discovery of what are now called Haydon spectra. It should be emphasized especially that in the work of Alexandroff on the homological theory of compacta and in the works of Pontryagin and A. Weil on compact groups all of the results are natural extensions of the results in the metric case. Mardešić’s theorem [S. Mardešić, Ill. J. Math. 4, 278-291 (1960; Zbl 0094.16902)] also has, to some extent, a “countable” counterpart – the theorem of H. Freudenthal [Compositio Math. 4, 145-234 (1937; Zbl 0016.28001)] which states that every metrizable compactum is the limit of an inverse sequence, consisting of polyhedra of the same dimension. In this sense Haydon’s method has an “uncountable” nature. It is based on a special spectral construction whose roots are in a simple set-theoretical principle known as “the effect of uncountability”. This principle, extracted in its present form by E. V. Shchepin [cf. Russ. Math. Surv. 36, No. 3, 1-71 (1981); translation from Usp. Mat. Nauk 36, No. 3(219), 3-62 (1981; Zbl 0463.54009)], serves as the foundation for the spectral theorem. This theorem is perhaps the most powerful tool of the whole theory. It states the following: if the limit spaces of two uncountable inverse spectra (with some additional non-restrictive properties) are homeomorphic, then these spectra contain isomorphic cofinal subspectra. The spectral theorem has no analog for inverse sequences: for example, two inverse sequences, consisting of even- and odd-dimensional cubes, produce the same limit space – the Hilbert cube – but, of course, they contain no isomorphic subspectra whatsover. Theorems stating that ‘limit spaces of spectra, consisting of “good” spaces are “good” ’ were typical in spectral applications. The spectral theorem allows to obtain theorems stating that limit spaces of spectra, consisting of “bad” spaces or “bad” projections are “bad”.
Detailed discussion of various versions (including compact and non-compact cases) of the spectral theorem is presented in Chapter 1.
Manifold theory or, more generally, the theory of absolute extensors, is one of the major areas of modern topology where the spectral approach has already shown its full strength. The author discusses this theory in detail. He starts with metrizable infinite-dimensional manifolds. This subject is almost completely covered in three excellent books [C. Bessaga and A. Pełczyński, Selected topics in infinite-dimensional topology, Monogr. Mat., Warszawa 58 (1975; Zbl 0304.57001); T. A. Chapman, Lectures on Hilbert cube manifolds, Reg. Conf. Ser. Math. 28 (1976; Zbl 0347.57005); J. van Mill, Infinite-dimensional topology, North-Holland Math. Libr. 43 (1989; Zbl 0663.57001)].
In Chapter 2 an updated survey of this theory is presented. The reader can find here proofs of several statements of \(Q\)-manifold and \(\ell_2\)-manifold theories. The author begins with an introduction of the concept of (strong) \(Z\)-sets and discusses all major results of these theories including H. Toruńczyk’s characterization theorems of the model spaces: countable infinite powers of the closed unit interval [Fundam. Math. 106, 31-40 (1980; Zbl 0346.57004)] and the real line [ibid. 111, 247-262 (1981; Zbl 0468.57015)].
Chapter 3 contains a solution of the well-known and long standing problem concerning coincidence of Lebesgue and integral cohomological dimensions.
Chapter 4 can be considered as an independent introduction to Menger manifold theory. The author presents a detailed discussion of this theory including the basic geometric constructions. He also presents examples of \(n\)-soft maps of \((n+1)\)-dimensional Menger compacta onto the Hilbert cube. These maps, constructions of which are performed using the spectral techniques, are used both in Menger manifold theory and later in the general theory of absolute extensors.
In Chapter 5 the author illustrates the close ties between Menger and Nöbeling spaces by showing how Nöbeling spaces can be identified with the so-called pseudo-interiors of Menger compacta. Using the spectral approach an \(n\)-soft map of \(n\)-dimensional Nöbeling space onto the separable Hilbert space is constructed.
In Chapters 6 and 7 the author develops the general theory of absolute extensors in dimension \(n\), \(n\in\omega\). The spectral theorem plays a crucial role in almost every statement here. As the culmination of this theory its two major elements are presented: the topological characterizations of uncountable powers of the closed unit interval and of the real line (originally done by E. V. Shchepin [Sov. Math., Dokl. 20, 511-515 (1979); translation from Dokl. Akad. Nauk SSSR 246, 551-554 (1979; Zbl 0431.54020)] and by the author [Math. USSR, Izv. 28, 151-174 (1987); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50, No. 1, 156-180 (1986; Zbl 0603.54018)], respectively). Coupled with the corresponding results of Toruńczyk the author therefore obtains topological characterizations of all infinite products of the closed unit interval and the real line.
Chapter 8 includes applications: topological characterizations of uncountable powers of the Cantor cube and of the separable Baire space; spectral representations of topological groups (including a simple proof of the existence of Lie series); topological characterization of locally convex linear topological spaces that are homeomorphic to the powers of the real line (generalization of the classical result of Anderson-Kadec); the spectral theorem in shape category; actions of non-metrizable topological groups and the structure of fixed point sets in non-metrizable manifolds; the spectral theorem for Baire maps (isomorphisms); connections with direct spectra and \(\Sigma\)-products.
Each section is ended with historical and bibliographical notes. Throughout the text the reader can find discussions of several unsolved problems.
The Bibliography comprises 326 titles.

MSC:

54-02 Research exposition (monographs, survey articles) pertaining to general topology
54B35 Spectra in general topology
54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
54F45 Dimension theory in general topology
55M10 Dimension theory in algebraic topology
57N20 Topology of infinite-dimensional manifolds
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