Projection spectrum

A family of simplicial complexes $ \{ {N _ \alpha } : {\alpha \in A } \} $( cf. Simplicial complex) indexed by a directed set $ ( A , > ) $ such that for every pair of indices $ \alpha , \alpha ^ \prime \in A $ for which $ \alpha ^ \prime > \alpha $ a simplicial mapping (projection) $ \pi _ \alpha ^ {\alpha ^ \prime } $ is defined from the complex $ N _ {\alpha ^ \prime } $ onto the complex $ N _ \alpha $. It is also required that $ \pi _ \alpha ^ {\alpha ^ {\prime\prime} } = \pi _ \alpha ^ {\alpha ^ \prime } \pi _ {\alpha ^ \prime } ^ {\alpha ^ {\prime\prime} } $ when $ \alpha ^ {\prime\prime} > \alpha ^ \prime > \alpha $( transitivity condition). Then it is said that a projection spectrum $ S = \{ N _ \alpha , \pi _ \alpha ^ {\alpha ^ \prime } , A \} $, or simply $ S = \{ N _ \alpha , \pi _ \alpha ^ {\alpha ^ \prime } \} $, is given. This concept is due to P.S. Aleksandrov (see [2]); it is essentially equivalent to the general concept of an inverse system, or an inverse spectrum (see System (in a category)). Indeed, every complex $ N _ \alpha $ naturally gives rise to the partially ordered set of simplices of this complex, and hence to a topological $ T _ {0} $- space $ N _ \alpha $. The projections $ \pi _ \alpha ^ {\alpha ^ \prime } $ then become continuous mappings. Conversely, if $ \{ N _ \alpha , \pi _ \alpha ^ {\alpha ^ \prime } \} $ is an inverse system of topological $ T _ {0} $- spaces and continuous projections $ \pi _ \alpha ^ {\alpha ^ \prime } $, then each $ T _ {0} $- space $ N _ \alpha $ naturally turns into a partially ordered set, and this partially ordered set is realized in the form of the simplicial complex $ N _ \alpha $. Here the continuous projections $ \pi _ \alpha ^ {\alpha ^ \prime } $ become simplicial mappings.

The concepts of a "projection spectrum" (and hence of an inverse system of spaces) and of a nerve of a system of sets (see below) have influenced the development of topology. After their introduction it became possible to speak about a theory of approximation of complicated topological and algebraic-topological objects by simpler ones.

If for every $ \alpha \in A $ the complex $ N _ \alpha $ is finite, then the spectrum $ S = \{ N _ \alpha , \pi _ \alpha ^ {\alpha ^ \prime } \} $ is called a finite projection spectrum. With each projection spectrum $ S = \{ N _ \alpha , \pi _ \alpha ^ {\alpha ^ \prime } \} $ the following concepts are associated. Any collection $ \xi = \{ {t _ \alpha } : {\alpha \in A } \} $ of simplices, one from every complex $ N _ \alpha $ of the spectrum $ S $, is called a thread of this spectrum if for $ \alpha ^ \prime > \alpha $ always $ \pi _ \alpha ^ {\alpha ^ \prime } t _ {\alpha ^ \prime } = t _ \alpha $, where $ t _ \alpha , t _ {\alpha ^ \prime } \in \xi $. The set $ \overline{S}\; $ of all threads with the topology whose base consists of the sets of the form $ O t _ {\alpha _ {0} } = \{ {\xi ^ \prime \in S } : {t _ {\alpha _ {0} } ^ \prime \leq t _ {\alpha _ {0} } } \} $, where $ \alpha _ {0} \in A $, $ t _ {\alpha _ {0} } \in N _ {\alpha _ {0} } $ are arbitrary and $ t _ {\alpha _ {0} } ^ \prime < t _ {\alpha _ {0} } $ means that the simplex $ t _ {\alpha _ {0} } ^ \prime $ of the thread $ \xi ^ \prime $ in the complex $ N _ {\alpha _ {0} } $ is a face of the simplex $ t _ {\alpha _ {0} } $, is called the complete limit of the spectrum $ S $. The same topology will be obtained by inducing on $ \overline{S}\; $ the topology of the Tikhonov product $ \Pi \{ { {\mathcal N} _ \alpha } : {\alpha \in A } \} $, where $ {\mathcal N} _ \alpha $ is the topological $ T _ {0} $- space corresponding to the complex $ N _ \alpha $. A thread $ \xi ^ \prime = \{ t _ \alpha ^ \prime \} $ is ambient to a thread $ \xi = \{ t _ \alpha \} $ if for every $ \alpha \in A $ one has $ t _ \alpha ^ \prime \geq t _ \alpha $. A thread $ \xi $ is called maximal (respectively, minimal) if there is no thread different from $ \xi $ which is ambient to $ \xi $( respectively, to which $ \xi $ is ambient). The subspace of the complete limit space $ \overline{S}\; $ of the spectrum $ S $ consisting of all the maximal (minimal) threads is called the upper (lower) limit of the spectrum $ S $. The complete limit $ \overline{S}\; $ is a semi-regular $ T _ {0} $- space, while the upper limit $ \widehat{S} $ and the lower limit $ \check{S} $ are $ T _ {1} $- spaces. If $ S $ is a finite projection spectrum, then $ \overline{S}\; $, $ \widehat{S} $ and $ \check{S} $ are compact spaces.

At the foundation of the entire theory of approximation of topological spaces by polyhedra, or more precisely by simplicial complexes, lies the concept, introduced by Aleksandrov (see [1]), of the nerve of a system of sets. The nerve of a given system $ \alpha $ of sets is defined to be the simplicial complex $ N _ \alpha $ whose vertices are in one-to-one correspondence with the elements of the system $ \alpha $ and a set of vertices determines a simplex of the complex $ N _ \alpha $ if and only if the sets of the system $ \alpha $ corresponding to those vertices have non-empty intersection.

It is more convenient to consider so-called canonical coverings of a space $ X $. A locally finite (finite) covering $ \alpha $ of the space $ X $ is called canonical if its elements are (closed) canonical sets (in another terminology, regular closed sets, cf. Canonical set) with disjoint interiors. For two canonical coverings $ \alpha ^ \prime , \alpha $ of the space $ X $, if $ \alpha ^ \prime $ follows $ \alpha $, i.e. if $ \alpha ^ \prime $ is a refinement of $ \alpha $( in this case $ \alpha ^ \prime > \alpha $), then the natural simplicial mapping $ \pi _ \alpha ^ {\alpha ^ \prime } $( the projection) of the nerve $ N _ {\alpha ^ \prime } $ onto the nerve $ N _ \alpha $ is defined; it is given by assigning to each element $ A ^ {\alpha ^ \prime } $ of $ \alpha ^ \prime $ that unique element $ A ^ \alpha $ of $ \alpha $ for which $ A ^ \alpha \supset A ^ {\alpha ^ \prime } $. Let $ \mathfrak A ( X) $( respectively, $ \mathfrak A _ {0} ( X) $) denote the collection of all locally finite (finite) canonical coverings of the space $ X $. For every $ \alpha \in \mathfrak A ( X) $( respectively, $ \alpha \in \mathfrak A _ {0} ( X) $), the nerve $ N _ \alpha $ of $ \alpha $ is considered. If $ \alpha ^ \prime > \alpha $, then a simplicial mapping $ \pi _ \alpha ^ {\alpha ^ \prime } : N _ {\alpha ^ \prime } \rightarrow N _ \alpha $ is defined. The projection spectrum $ S = \{ N _ \alpha , \pi _ \alpha ^ {\alpha ^ \prime } \} $ thus obtained is called the complete (respectively, finite) projection spectrum of the topological space $ X $. Aleksandrov [2] proved in 1928 that every metric ( $ n $- dimensional) compact space is the upper limit of an ( $ n $- dimensional) finite projection spectrum over a countable set of indices. A.G. Kurosh proved in 1934 that every compactum is the upper limit of its finite projection spectrum. In 1961, V.I. Ponomarev proved that every paracompactum is the upper limit of its complete projection spectrum, that is, the spectrum constructed over the set $ \mathfrak A ( X) $ of all locally finite canonical coverings of the space $ X $. Ponomarev has introduced the concept of a relaxation of a simplicial complex $ K $, by which he means any closed subcomplex $ K ^ \prime \subset K $ containing all vertices of the complex $ K $. The zero-dimensional complex consisting of all vertices of the complex $ K $ is called its total relaxation (or skeleton). By replacing all the complexes of a given projection spectrum by their (total) relaxations while preserving the projections, one obtains the (total) relaxation of the spectrum. The investigation of irreducible perfect mappings of paracompacta reduces to the study of the relaxations of their complete projection spectra. Here the limit of the total relaxation of the complete projection spectrum of a paracompactum $ X $ is the so-called absolute $ \dot{X} $ of $ X $, and the limit of the total relaxation of the finite projection spectrum of any regular space is the Stone—Čech compactification $ \beta \dot{X} $ of the absolute $ \dot{X} $ of that regular space. Every finite abstract projection spectrum is equivalent to the spectrum over a directed refining set of finite canonical coverings of some semi-regular compact $ T _ {0} $- space, that is, it is obtained from this spectrum by means of finitely many of the following operations: 1) replacement of a spectrum by an isomorphic spectrum; 2) replacement of a spectrum by a cofinal part of it; 3) replacement of a spectrum by a spectrum containing the given one as a cofinal part (Zaitsev's theorem).

The concepts of a nerve and a projection spectrum provided the means for reducing the properties of general spaces, first of all, paracompacta, compacta and metric compacta, to the properties of complexes and their simplicial mappings. This made it possible to define and to study homology and cohomology invariants not only of polyhedra but of general spaces (see Aleksandrov–Čech homology and cohomology; Spectral homology). All this has led to the synthesis of geometric and set-theoretic ideas in topology.

References

[1]	P.S. Aleksandrov, "Une définition des nombres de Betti pour un ensemble fermé quelconque" C.R. Acad. Sci. Paris , 184 (1927) pp. 317–319
[2]	P.S. Aleksandrov, "Untersuchungen über Gestalt und Lage abgeschlossener Mengen beliebiger Dimension" Ann. of Math. , 30 (1929) pp. 101–187
[3]	P.S. Aleksandrov, "On the concept of space in topology" Uspekhi Mat. Nauk , 2 : 1 (1947) pp. 5–57 (In Russian)
[4]	P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[5]	P.S. Aleksandrov, V.I. Ponomarev, "Projection spectra" J. Novák (ed.) , General topology and its relations to modern analysis and algebra , 2 , Czechoslovak. Acad. Sci. (1967) pp. 25–30 (In Russian)
[6]	P.S. Aleksandrov, V.V. Fedorchuk, "The main aspects in the development of set-theoretical topology" Russian Math. Surveys , 33 : 3 (1978) pp. 1–53 Uspekhi Mat. Nauk , 33 : 3 (1978) pp. 3–48
[7]	V.I. Ponomarev, "Projective spectra and continuous mappings of paracompacta" Transl. Amer. Math. Soc. (2) , 39 (1964) pp. 133–164 Mat. Sb. , 60 : 1 (1963) pp. 89–119
[8]	V.I. Zaitsev, "Projection spectra" Trans. Moscow Math. Soc. , 27 (1972) pp. 135–200 Trudy Moskov. Mat. Obshch. , 27 (1972) pp. 129–119

Comments

In the West, the concept of a projection spectrum is generally considered to have mainly historical importance, as the first version of the crucial concept of inverse system. For instance, the fundamental continuity axiom of abstract homology (cf. Steenrod–Eilenberg axioms) is stated for complete limits of inverse systems of spaces. Picking out the upper limit succeeds, as indicated above, in representing a space by means of suitable nerves of coverings, but it combines badly with other topological constructions.

In the 1950-s and 1960-s several topologists, principally S. Mardešić [a2] and B.A. Pasynkov [a5], discovered some undesirable features of the representation of $ n $- dimensional spaces as inverse limits of systems of $ n $- dimensional polyhedra. The keynote of this work is sounded by Mardešić's theorem: Every compact Hausdorff space of covering dimension $ n $ is an inverse limit of inverse limits of $ n $- dimensional finite polyhedra. Recently Mardešić and collaborators have successfully explained some of the principal peculiarities, using the concept of an approximate inverse limit. The approximate inverse limit of an inverse system coincides with its complete limit; but one also permits approximate inverse systems. See [a3], [a4].

References

[a1]	S. Eilenberg, N.E. Steenrod, "Foundations of algebraic topology" , Princeton Univ. Press (1966)
[a2]	S. Mardešić, "On covering dimension and inverse limits of compact spaces" Ill. J. Math. , 4 (1960) pp. 278–291
[a3]	S. Mardešić, L. Rubin, "Approximate inverse systems of compacta and covering dimension" Pacific J. Math. , 138 (1989) pp. 129–144
[a4]	S. Mardešić, J. Segal, "Stability of almost commutative inverse systems for compacta" Topology Appl. , 31 (1989) pp. 285–299
[a5]	B.A. Pasynkov, "On spectra and dimension of topological spaces" Mat. Sb. , 57 (99) (1962) pp. 449–476 (In Russian)