Topology Atlas Document # taic-43 Topology Atlas Invited Contributions vol. 6 issue 1 (2001) p. 7-73

Cohomological dimension theory of compact metric spaces

A.N. Dranishnikov

Department of Mathematics
University of Florida
444 Little Hall
Gainesville, FL 32611-8105

dranish@math.ufl.edu
http://www.math.ufl.edu/~dranish/

Formats

Contents

  1. Introduction.
  2. General properties of cohomological dimension.
  3. Bockstein Theory.
  4. Cohomological dimension of product.
  5. Dimension type algebra.
  6. Realization theorem.
  7. Test spaces.
  8. Infinite dimensional comapacta with finite cohomological dimension.
  9. Resolution theorems.
  10. Resolutions preserving cohomological dimension.
  11. Classifying spaces for cohomological dimension.
  12. Imbedding and approximation.
  13. Cohomological dimension of ANRs.

Introduction

The cohomological dimension theory has connections with many different areas of mathematics: dimension theory, topology of manifolds, group theory, functional rings and others. It was founded by P.S. Alexandroff in late 20's. Many famous topologists have contributed to the theory. Among them are Hopf, Pontryagin, Bockstein, Borsuk, Dyer, Boltyanskij, Kodama, Kuzminov, Sitnikov.

There are only few introductory and survey texts on the theory. The book by Alexandroff `Introduction to homological dimension theory and general combinatorial topology' is written in old fashion language and hardly readable. There are surveys by Kodama (Appendix in [K. Nagami, Dimension theory, Academic Press, 1970]) and by Kuzminov [Homological dimension theory, Russian Mathematical Surveys 23 (1968), 1-45]. The first part of Kuzminov paper is devoted to compact metric spaces and is an excellent reading. We don't consider noncompact spaces in this paper, since the cohomological dimension of noncompact spaces behaves differently and is not completely developed. A very special case of the cohomological dimension theory is the case of integer coefficients. An excellent survey on this case was written by Walsh [Dimension, cohomological dimension, and cell-like mappings, Lecture Notes in Math. 870, Springer, 1981, pp. 105-118] where a detailed proof of the Edwards resolution theorem first was published. In 1988, twenty years after Kuzminov's survey I wrote a sequel to that [Homological dimension theory, Russian Mathematical Surveys 43 (1988), 11-63]. Since then ten years passed, new results appeared and a new understanding of the old results ripened. So time came for an update survey. A new compressed survey was given by Dydak [Cohomological dimension theory, preprint, 1997] where the main applications of the cohomological dimensions are discussed. Here we present a detailed introductory survey of the theory.

This survey and Dydak's have the same origin. They appeared as the notes to our joint book that we planned to write [A.N. Dranishnikov, J. Dydak and J. Walsh, Cohomological dimension theory with applications, preprint, 1992]. We still have a hope that someday we will accomplish that.

In this paper we assume that the reader is familiar with basic elements of the homotopy theory, with homology and cohomology theories, including the Cech cohomology, the Steenrod homology and extraordinary (co)homologies. Some knowledge in the dimension theory and the theory of absolute neighborhood retracts will be useful. Also we don't discuss here any applications of the cohomological dimension theory even to the dimension theory. Interested reader can find a discussion of some applications in Dydak's survey mentioned above.

I am thankful to Topology Atlas for inviting me to write this survey. I am also thankful to NSF, DMS-9971709, for the support.

Published: March 29, 2001.