This is a comprehensive introduction into the method of inverse spectra - a powerful method successfully employed in various branches of topology.
The notion of an inverse sequence and its limits, first appeared in the well-known memoir by Alexandrov where a special case of inverse spectra - the so-called projective spectra - were considered. The concept of an inverse spectrum in its present form was first introduced by Lefschetz. Meanwhile, Freudental, had introduced the notion of a morphism of inverse spectra. The foundations of the entire method of inverse spectra were laid down in these basic works.
Subsequently, inverse spectra began to be widely studied and applied, not only in the various major branches of topology, but also in functional analysis and algebra. This is not surprising considering the categorical nature of inverse spectra and the extraordinary power of the related techniques.
Updated surveys (including proofs of several statements) of the Hilbert cube and Hilbert space manifold theories are included in the book. Recent developments of the Menger and Nöbeling manifold theories are also presented.
This work significantly extends and updates the author's previously published book and has been completely rewritten in order to incorporate new developments in the field.
Inverse Spectra, 1st Edition
Preface. Inverse Spectra.
Preliminary information. Definitions and elementary properties of inverse spectra. Factorizing spectra and the spectral theorem. Infinite-Dimensional Manifolds.
Absolute extensors and absolute retracts. Z
-sets in AN R
- and I&ohgr;
-manifolds. Topology of R&ohgr;
- and I&ohgr;
-manifolds. Incomplete manifolds. Cohomological Dimension.
Cohomological dimension. Cell-like mappings raising dimension. Universal space for cohomological dimension. Menger Manifolds.
General theory. n
-soft mappings of compacta, raising dimension. n
-soft mappings of Polish spaces, raising dimension. Further properties of Menger manifolds. Homeomorphism groups. &ohgr;-soft map of &sgr; onto &Sgr;. Nöbeling Manifolds.
-universal spaces. Pseudo-interiors and pseuod-boundaries of Menger compacta. Geometric pseudo-boundaries. Equivalence of categorical and geometric pseudo-interiors. Equivalence of the Nöbeling space and the pseudo-interior of &mgr;n
. Further properties of Nöbeling spaces. Open subspaces of Nöbeling spaces. General Theory of Absolute Extensors in Dimension n and n-soft Mappings. AN E
)-spaces and n
-soft mappings. Morphisms of spectra and square diagrams. Spectral characterizations of n
-soft mappings. Further properties of AE
(0)-spaces. Strongly universal spaces. Topology of Non-Metrizable Manifolds.
Non-metrizable manifolds. Topological characterization of I&tgr;
-manifolds. Topological characterization of R&tgr;
-manifolds. Trivial bundles. Applications.
Uncountable powers of countable discrete spaces. Spectral representations of topological groups. Locally convex linear topological spaces. Shape properties of non-metrizable compacta. Fixed point sets of Tychonov cubes. Compact groups and fixed point sets. Group actions. Baire isomorphisms. Double spectra. Skeletoids in Tychonov cubes. Bibliography. Subject Index.