##
**Arithmetic of dimension theory.**
*(English.
Russian original)*
Zbl 0967.55001

Russ. Math. Surv. 53, No. 5, 975-1069 (1998); translation from Usp. Mat. Nauk. 53, No. 5, 115-212 (1998).

This survey, involving certain aspects of dimension and extension theory, consists of four parts: an Introduction and Chapters I–III. At 95 pages, it is more like a short book, which may make it a formidable challenge to a casual reader, especially to one not totally versed in all the aspects covered by it. It certainly presents a major challenge for a reviewer, since the author has been obscure about the goals of this paper. Therefore we apologize in advance for any assumptions we seem to make or any misconceptions we have concerning the overall drift of this survey. We remark also that we are working only with the English translation of the original Russian, so it is possible that we are not getting the full meaning intended in the latter.

The Introduction appears to be in part a history of dimension theory, but mainly it is a lead-in for the three main chapters, an attempt to focus the reader on a certain trail which can be traced in a specific direction. It involves a fast leap over more than half a century of the development of this subject. A reader wanting to know more about this history would be well-advised to study several sources, possibly beginning with the classic work [Dimension theory (1948; Zbl 0036.12501)] by W. Hurewicz and H. Wallman. Here at least one could learn elementary facts about the dimension function dim for separable, metrizable spaces. Another source, which contains many remarks about the early history of dimension theory would be the book [Theory of dimensions, finite and infinite, Sigma Ser. Pure Math. 10 (1995; Zbl 0872.54002)] by R. Engelking.

After the short historical portion, the author begins to focus on the topics of most interest to him. These, it seems to us, involve the area of homological dimension theory, introduced in a 1932 paper of P. S. Alexandroff [Math. Ann. 106, 161-238 (1932; Zbl 0004.07301)]. This subject was to evolve later into cohomological dimension theory, or, we should say, “theories,” \(\dim_G\), one for each abelian group \(G\). On the other hand, at least in the class of metrizable compacta (the important class of spaces for this paper) there is a globally unifying theory whose development began with work of M. Bokshtein (1950’s) and which has had more recent fulfillment via the efforts of A. Dranishnikov. Of course there were other advances in between and in conjunction with the work of these two researchers and many others.

Among the important developments for cohomological dimension was the Bokshtein Basis Theorem. This shows that for a given abelian group \(G\), in order to determine \(\dim _G X\) for a given metrizable compactum \(X\), it is sufficient to know \(\dim _H X\) for all \(H\) in the “Bokshtein Basis” of \(G\). The strength of this comes from the fact that there is a fixed, countable collection \(\mathcal B\) of abelian groups such that the Bokshtein Basis of any \(G\) lies in \(\mathcal B\). Moreover, there are criteria for determining for the given \(G\) which elements of \(\mathcal B\) are in its basis. Also, and significant for the direction of this survey, there is a list of “Bokshtein inequalities” which relate \(\dim _G X\) to \(\dim _H X\) whenever \(G\), \(H\in\mathcal B\), independently of choice of \(X\).

As is known, \(\dim (X\times Y)\leq \dim X+\dim Y\), but even for some compacta, that inequality can become strict. This is possible also when \(X=Y\). Cohomological dimension theory, on the other side, has proved to be more effective when considering the dimension of products. But, in some cases, the inequality can actually reverse. Much has been done to codify the dimension of a product under \(\dim _G\) for a given group \(G\).

The situation involving the dimension of a product has been associated with the unstable intersection problem. If two maps \(f\), \(g\) respectively of compacta \(X\) and \(Y\) into \(\mathbb R^n\) are given, then under what conditions is it possible to ensure that there are maps \(f'\), \(g'\), respectively approximating \(f\), \(g\) as closely as desired, in such a manner that \(f'(X)\cap f'(Y)=\emptyset\)? Indeed, we may also ask what is the minimum possible dimension of such an intersection if it cannot be made void.

In a sequence of events, related to the unstable intersection situation, first came a question of F. D. Ancel (incorrectly spelled Aczél on page 982 of this survey) concerning approximating maps into \(\mathbb R^{2n}\) by embeddings. The initial paper on this, by D. McCullough and L. Rubin [Fundam. Math. 116, 131-142 (1983; Zbl 0552.55003)], had an error which was discovered by J. Krasinkiewicz and K. Lorentz. It was the second (and correct) paper of D. McCullough and L. R. Rubin [ibid. 133, No. 3, 237-245 (1989; Zbl 0715.54026)], in which the important remediation was made, and which is considered to be the start of a series of papers relating the approximation by embeddings to the dimension of the product and ultimately to the unstable intersection problems. This part of the history, as posed on page 982 of this survey, has not been put correctly.

Let us mention, in passing, a couple of other places where the results seem not to be stated correctly. The Corollary on page 983 probably should require that \(\dim X<\infty\). On page 984, the completion theorem, ascribed to Olszewski, is not stated with the applicable constraints in the hypothesis. We suggest that the reader interested in precision, should consult the sources in these cases as well as the others.

Outside the lengthy Introduction are the three main chapters of the paper. The first one is entitled, Algebra. It is designed to bring to the attention of the reader the kinds of facts needed for the tensor classification of abelian groups, certain Künneth algebras, and then finally what is called the “code lattice”. Apparently there is a related code algebra whose computations are connected with the theory of intersections of compacta (see page 991).

Chapter II deals with the theory of extensional dimension. The basic idea for this is the notion \(X\tau K\) where \(X\) is a space and \(K\) is the polyhedron of a simplicial complex. This notation means that for each closed subset \(A\) of \(X\) and map \(f:A\rightarrow K\), there exists a map \(F:X\rightarrow K\) which is an extension of \(f\). It is known that \(X\tau S^n\) is equivalent to \(\dim X\leq n\), and \(X\tau K(G,n)\) is equivalent to \(\dim _G X\leq n\). A whole subject called extension theory has developed around this concept. One may consult A. N. Dranishnikov’s 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)] for some of the rudiments. Chapter II attempts to provide a survey of some of the principal notions in extension theory.

Finally, Chapter III is the place where all the connections of the instruction given in Chapters I and II are applied to give answers to many questions concerning the unstable intersection problem. It is shown how the theory of cohomological dimension is related, how products come into play, and why the coding can carry the relevant information.

The Introduction appears to be in part a history of dimension theory, but mainly it is a lead-in for the three main chapters, an attempt to focus the reader on a certain trail which can be traced in a specific direction. It involves a fast leap over more than half a century of the development of this subject. A reader wanting to know more about this history would be well-advised to study several sources, possibly beginning with the classic work [Dimension theory (1948; Zbl 0036.12501)] by W. Hurewicz and H. Wallman. Here at least one could learn elementary facts about the dimension function dim for separable, metrizable spaces. Another source, which contains many remarks about the early history of dimension theory would be the book [Theory of dimensions, finite and infinite, Sigma Ser. Pure Math. 10 (1995; Zbl 0872.54002)] by R. Engelking.

After the short historical portion, the author begins to focus on the topics of most interest to him. These, it seems to us, involve the area of homological dimension theory, introduced in a 1932 paper of P. S. Alexandroff [Math. Ann. 106, 161-238 (1932; Zbl 0004.07301)]. This subject was to evolve later into cohomological dimension theory, or, we should say, “theories,” \(\dim_G\), one for each abelian group \(G\). On the other hand, at least in the class of metrizable compacta (the important class of spaces for this paper) there is a globally unifying theory whose development began with work of M. Bokshtein (1950’s) and which has had more recent fulfillment via the efforts of A. Dranishnikov. Of course there were other advances in between and in conjunction with the work of these two researchers and many others.

Among the important developments for cohomological dimension was the Bokshtein Basis Theorem. This shows that for a given abelian group \(G\), in order to determine \(\dim _G X\) for a given metrizable compactum \(X\), it is sufficient to know \(\dim _H X\) for all \(H\) in the “Bokshtein Basis” of \(G\). The strength of this comes from the fact that there is a fixed, countable collection \(\mathcal B\) of abelian groups such that the Bokshtein Basis of any \(G\) lies in \(\mathcal B\). Moreover, there are criteria for determining for the given \(G\) which elements of \(\mathcal B\) are in its basis. Also, and significant for the direction of this survey, there is a list of “Bokshtein inequalities” which relate \(\dim _G X\) to \(\dim _H X\) whenever \(G\), \(H\in\mathcal B\), independently of choice of \(X\).

As is known, \(\dim (X\times Y)\leq \dim X+\dim Y\), but even for some compacta, that inequality can become strict. This is possible also when \(X=Y\). Cohomological dimension theory, on the other side, has proved to be more effective when considering the dimension of products. But, in some cases, the inequality can actually reverse. Much has been done to codify the dimension of a product under \(\dim _G\) for a given group \(G\).

The situation involving the dimension of a product has been associated with the unstable intersection problem. If two maps \(f\), \(g\) respectively of compacta \(X\) and \(Y\) into \(\mathbb R^n\) are given, then under what conditions is it possible to ensure that there are maps \(f'\), \(g'\), respectively approximating \(f\), \(g\) as closely as desired, in such a manner that \(f'(X)\cap f'(Y)=\emptyset\)? Indeed, we may also ask what is the minimum possible dimension of such an intersection if it cannot be made void.

In a sequence of events, related to the unstable intersection situation, first came a question of F. D. Ancel (incorrectly spelled Aczél on page 982 of this survey) concerning approximating maps into \(\mathbb R^{2n}\) by embeddings. The initial paper on this, by D. McCullough and L. Rubin [Fundam. Math. 116, 131-142 (1983; Zbl 0552.55003)], had an error which was discovered by J. Krasinkiewicz and K. Lorentz. It was the second (and correct) paper of D. McCullough and L. R. Rubin [ibid. 133, No. 3, 237-245 (1989; Zbl 0715.54026)], in which the important remediation was made, and which is considered to be the start of a series of papers relating the approximation by embeddings to the dimension of the product and ultimately to the unstable intersection problems. This part of the history, as posed on page 982 of this survey, has not been put correctly.

Let us mention, in passing, a couple of other places where the results seem not to be stated correctly. The Corollary on page 983 probably should require that \(\dim X<\infty\). On page 984, the completion theorem, ascribed to Olszewski, is not stated with the applicable constraints in the hypothesis. We suggest that the reader interested in precision, should consult the sources in these cases as well as the others.

Outside the lengthy Introduction are the three main chapters of the paper. The first one is entitled, Algebra. It is designed to bring to the attention of the reader the kinds of facts needed for the tensor classification of abelian groups, certain Künneth algebras, and then finally what is called the “code lattice”. Apparently there is a related code algebra whose computations are connected with the theory of intersections of compacta (see page 991).

Chapter II deals with the theory of extensional dimension. The basic idea for this is the notion \(X\tau K\) where \(X\) is a space and \(K\) is the polyhedron of a simplicial complex. This notation means that for each closed subset \(A\) of \(X\) and map \(f:A\rightarrow K\), there exists a map \(F:X\rightarrow K\) which is an extension of \(f\). It is known that \(X\tau S^n\) is equivalent to \(\dim X\leq n\), and \(X\tau K(G,n)\) is equivalent to \(\dim _G X\leq n\). A whole subject called extension theory has developed around this concept. One may consult A. N. Dranishnikov’s 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)] for some of the rudiments. Chapter II attempts to provide a survey of some of the principal notions in extension theory.

Finally, Chapter III is the place where all the connections of the instruction given in Chapters I and II are applied to give answers to many questions concerning the unstable intersection problem. It is shown how the theory of cohomological dimension is related, how products come into play, and why the coding can carry the relevant information.

Reviewer: L.R.Rubin (Norman)

### MSC:

55M10 | Dimension theory in algebraic topology |

54F45 | Dimension theory in general topology |

54D35 | Extensions of spaces (compactifications, supercompactifications, completions, etc.) |

57N17 | Topology of topological vector spaces |

20K99 | Abelian groups |

54C35 | Function spaces in general topology |