Katsuro Sakai has written an important and well-timed monograph on a pair of subjects in algebraic (and general) topology that have seen a lot of relatively recent activity: dimension theory and the theory of retracts. He proffers two exemplars of this recent activity as the raison d’être for his book, namely, the resolution in the late 1980s of the Alexandroff problem and Cauty’s 1994 counterexample to the conjecture that any metrizable topological vector space should be an absolute retract. The Alexandroff problem asks for an infinite dimensional space of finite cohomological dimension, and it was Dranishnikov who provided a construction of an infinite dimensional compact metrizable space with finite cohomological dimension, in 1988. Sakai states that “[t]hese results are discussed in the latter half of the final chapter and provide an understanding of how deeply [dimension theory and the theory of retracts] are related to each other.”
What about the foregoing half-dozen chapters? Well, Sakai’s book has served him as the text for his graduate course at the University of Tsukuba. The reader clearly should not come to this book as a topology neophyte; such standards as Munkres’ and Dugundji’s books, both titled Topology, would serve well as prerequisites, but Sakai is quite mindful of his charges’ mathematical youthfulness. He notes, too, that “[e]xcept for the [aforementioned] latter half of the final chapter, th[e] book is self-contained.” So it is that chapters 2–6 are filled with material proper to a solid graduate course in topology, but with a particular set of goals in mind, i.e. dimension theory and retracts. This adds a special and desirable feature to the book, in that it is obviously well-situated to inaugurate a fledgling into the indicated pet topics of the author, making for a good prelude to solid research.
It should be noted, of course, that linear spaces are perhaps more heavily represented than might otherwise be the case, but this is clearly a virtue in the context of any mathematical education, and it is well-timed. Additionally, there is a great focus trained on simplicial methods, but one can again argue that this is a virtue.
The first chapter is devoted to preliminaries, the second to metrization and paracompactness — we meet Tietze and Stone and get to play with partitions of unity. Next, in the third chapter, linear spaces appear, as well as convexity (Hahn-Banach, the closed graph theorem and the open mapping theorem). The fourth chapter is on simplicial methods (and is very thorough as well as being well-written; indeed, this can be said about the entire book). I find Sakai’s treatment of nerves of open covers particularly evocative. I first learned it in the undeniable classic Dimension Theory by Hurewicz and Wallman, but it is hugely valuable to see it presented in a modern idiom with Sakai’s well-chosen illustrations.
The fifth chapter gives us dimensions of spaces, starting with Brouwer and finishing with an appendix on Hahn-Mazurkiewicz. Chapter six is devoted to “retracts and extensors” and clearly revs the reader up for the promised gems in the seventh and final chapter. Of course retracts are of central importance to topology in an even broader sense, so the importance of this chapter in and of itself cannot be overstated. Homotopy extension appears relatively early on, and (local) n-connectedness appears toward the chapter’s end. And then it’s on to chapter seven.
Geometric Aspects of General Topology is a well-written, nicely-illustrated, scholarly text, and should serve as a solid counterweight, so to speak, to the dominance of algebraic topology in this part of the graduate curriculum. It is also, as already indicated, a springboard to further work on dimension and retract theory. It’s a fine book.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.