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Cohomological dimension theory. (English) Zbl 0992.55001
Daverman, R. J. (ed.) et al., Handbook of geometric topology. Amsterdam: Elsevier. 423-470 (2002).
The theory of cohomological dimension began with work of P. S. Alexandroff in the late 1920’s and the early 1930’s. The cohomological dimension of a topological space is a number which assigns to the space some measure of its “thickness” and which agrees with our sense of dimension for the standard Euclidean spaces. At first it was defined only with respect to the group of integers; and Alexandroff proved that, at least for finite-dimensional compact metrizable spaces, this notion of dimension agreed with the previous notions of large and small inductive dimension. Eventually, the theory of cohomological dimension split into an infinite set of theories of dimension, one for each abelian group $G$, and the symbol dim$_G X$ is used to denote this value for a given space $X$. The purpose of this article is to explore that theory and to explain some of the directions of research which have developed because of its influence. Two other sources of information on this topic which may serve the reader are a preprint of {\it A. N. Dranishnikov} entitled, “Cohomological dimension theory of compact metric spaces” and an article by {\it E. V. Shchepin} [Russ. Math. Surv. 53, No. 5, 975-1069 (1998); translation from Usp. Mat. Nauk 53, No. 5, 115-212 (1998; Zbl 0967.55001)]. Some milestones pointed out are the application of Pontryagin of cohomological dimension theory to the proof of the existence of a metrizable compactum of dimension 2 whose product with itself has dimension 3, the proof by Bockstein that there is a countable basis (called the Bockstein groups) which can be used to detect the cohomological dimension of any compact Hausdorff space, the Edwards-Walsh resolution theorem, and the proof by A. Dranishnikov of the existence of a metrizable compactum which has infinite covering dimension and finite integral cohomological dimension. The author points out, and the reviewer concurs, that the work [Lect. Notes Math. 870, 105-118 (1981; Zbl 0474.55002)] by {\it J. J. Walsh}, was decisive in determining the direction in which the theory of cohomological dimension would proceed, and, indeed, providing the impetus for the emergence of extension theory. Not only did this work build a bridge connecting the question of the existence of cell-like dimension-raising maps [{\it L. R. Rubin}, Banach Cent. Publ. 18, 371-376 (1986; Zbl 0634.55001)] with the problem of whether dimension and integral cohomological dimension always agree, but also it laid out in plain terms that all the theories of dimension in question could be studied from the point of view of extensions of maps to certain types of complexes, the fundamental notion of extension theory. The latter term was coined and the fundamentals of the theory established by {\it A. N. Dranishnikov} in his seminal work [Russ. Acad. Sci., Sb., Math. 81, No. 2, 467-475 (1995); translation from Mat. Sb. 185, No. 4, 81-90 (1994; Zbl 0832.55001)]. This survey lays out the line of history leading from the beginning of the theories of dimension and cohomological dimension to intermediate stages of the development of cohomological dimension and finally to its impact on the development of extension theory and of its connections with other branches of topology. It provides a long, useful list of references, which when combined with those in the works of Dranishnikov and Shchepin mentioned above would be very helpful in guiding a researcher or an interested graduate student into the literature of the subject. A certain amount of caution should be used when employing the published results of the subjects mentioned above. For example, the reader can learn from the work “Cohomological dimension and acyclic resolutions” by {\it A. Koyama} and {\it K. Yokoi}, which is to appear in Topol. Appl., about some early mistakes in the proof of the existence of Edwards-Walsh complexes for certain groups, and can also by a very careful reading see that the original proof of the rational resolution theorem contains a gap. Only recently in work of {\it I. Ivansić} and {\it L. Rubin} (and then subsequently by {\it J. Dydak}) has a satisfactory proof of the existence of extension dimension for a major class of spaces, according to its original definition, been constructed. Finally, more results in the area have been placed into view during the last year, and of course have not been considered in this presentation. For the entire collection see [Zbl 0977.00029].

MSC:
55M10Dimension theory (algebraic topology)
54F45Dimension theory (general topology)
55-02Research monographs (algebraic topology)