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Higher-Dimensional Generalized Manifolds: Surgery and Constructions

Alberto Cavicchioli, Friedrich Hegenbarth, and Dušan Repovš
European Mathematical Society
Publication Date: 
Number of Pages: 
EMS Series of Lectures in Mathematics
[Reviewed by
Michael Berg
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Differential geometry is a very sexy subject. It has been so from the beginning, but its appeal has increased over the last few decades, for example due to dramatic developments in low dimensional topology and string theory. It is fair to say that just about all mathematicians need to know a decent sized chunk of this material regardless of their specialties. Happily, this subject, often bracketed with topology, is a major player in graduate programs, often a recommended, if not required, subject, and there are a number of excellent texts to aid one in getting off the ground and decently airborne in this area. I have had the pleasure of reviewing a number of good books along these lines in this column, among them the recent book by Clifford Taubes, and Loring Tu’s eminently accessible text, An Introduction to Manifolds. (I think that it would be a crime not to follow a thorough study of Tu’s book with an equally thorough study of the classic book Tu wrote together with his advisor, Raoul Bott, namely, Differential Forms in Algebraic Topology.) There are other ways to go, of course, but as the song goes, these are a few of my favorite things. In any case, before you have a go at the book under review, be sure that mainstream, orthodox differential geometry, along the preceding lines, is not alien to you…

The book under review certainly looks like it’s about differential geometry, too, but there’s more: the manifolds are “generalized” and topology is featured rather explicitly, with the tell-tale term “surgery” jumping out from the title page. Indeed, what’s with the surgery business? Well, the answer to these queries can be gleaned from the remarks on the book’s back cover:

Generalized manifolds … were introduced in the 1930s when topologists tried to detect topological manifolds among more general spaces … However, it soon became more important to study the category of generalized manifolds itself. A breakthrough was made in the 1990s, when several topologists discovered a systematic way of constructing higher-dimensional generalized manifolds, based on advanced surgery techniques.

All right, fair enough. What, then, is a generalized manifold, as opposed to just a garden variety manifold? Well, on p. 2 of the book under review we find that, by definition, a topological space X is a generalized manifold iff

(i) X is a Euclidean neighborhood retract [and] (ii) X is a homology n-manifold.

Actually, this is a more modern formulation of the notion, specifically defining a geometric generalized n-manifold; classically (before the 1950s) the definition asks for the following: (i) X has to be cohomologically locally connected, meaning that points admit neighborhoods that qualify as targets for trivializations of restrictions of cohomology (be it Čech or sheaf); (ii) X has to have finite cohomological dimension, meaning that for j uniformly large enough, cohomology with compact supports vanishes in dimension j for all open sets of X (take cohomology with coefficients in a PID, R, pre-given: we’re playing with generalized manifolds over R); finally, (iii) for certain homology sheaves the stalks are to be free R-modules of rank 1. To get more precise characterizations, crack the book.

So the handwriting on the wall is pretty clear: the reader had better know a good deal of manifold theory per se, as well as topology per se, particularly surgery theory, etc., and then more cohomology theory than the average bear knows. And there’s a lot more: just check the first pages of the book. Yes, it’s pretty specialized stuff, as is in fact already gleaned from the Preface’s first sentence:

These notes arose in the course of our studies of the systematic construction of higher-dimensional generalized manifolds given by Bryant, Ferry, Mio, and Weinberger…

Beware: heavy and specialized (but interesting) stuff.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

See the table of contents in pdf format.