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Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids

Ronald Brown, Philip J. Higgins, and Rafael Sivera
European Mathematical Society
Publication Date: 
Number of Pages: 
Tracts in Mathematics 15
[Reviewed by
Michael Berg
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As one myself, I heartily embrace the aphorism that a mathematician, regardless of what he may claim otherwise, is at heart a Platonist. Accordingly the following utterance by Alexander Grothendieck is undeniably encouraging: “Don’t be surprised by my supposed efficiency in digging out the right kind of notions — I have just been following, rather let myself be pulled ahead, by that very strong thread (roughly: understand non-commutative cohomology of topoi) which I kept trying to sell for ten or twenty years now, without anyone ready to ‘buy’ it, namely to do the work. So finally I got mad and decided to work out at least an outline by myself…” [My italics]. I note, too, that this disclosure by Grothendieck fits well with his famous and evocative metaphor of “the rising sea” to describe his approach to solving a mathematical problem: eventually the problem, symbolized by an island in the sea of surrounding mathematical methodology, is flooded by the water and disappears.

The quote above is from a letter Grothendieck apparently wrote to Ronald Brown, one of the three authors of the book under review, in 1983, and the topic of non-commutative cohomology is the theme of the book’s twelfth chapter. Wikipedia defines a topos as “a type of category that behaves like the category of sheaves of sets on a topological space,” and so we are actually on somewhat safe and familiar ground, as least as far as part of what the book is about.

But what about the three things mentioned on the front-cover? Filtered spaces, crossed complexes, and [!] cubical homotopy groupoids? Well, a filtered space is innocuous enough: on p.xxv, and again on p.211, we find that it’s a finite nested sequence of subspaces terminating in the named space. But then, on p. xxvi (and then on p.214), it gets stickier fast: a crossed complex is a left-infinite sequence of totally disconnected groupoids, with the last two being abelian, equipped with an action of the terminal (rightmost) groupoid on each of its predecessors, and otherwise mimicking the more prosaic notion of a chain complex in that the constituent morphisms satisfy dn-1dn=0 for n>2 (and there’s one more technical condition). Well, fair enough: there is still a lot of resonance here, especially if one bears in mind what a groupoid is, namely (cf. p.12), “a small category in which every arrow is invertible.” Evidently it’s all about homological algebra, apparently on anabolic steroids and accordingly developed to deal with a certain set of questions in algebraic topology.

All right, what kinds of questions, then? We let the authors speak for themselves: “The [book] project arose from the question formulated in about 1965 as to whether or not groupoids could be used in higher homotopy theory. Could one develop theories and applications of higher groupoids in a spirit similar to that of combinatorial group theory … thus continuing J. H. C. Whitehead’s project of ‘combinatorial homotopy’?” A propos, this parallel builds on the fact, presented also on p.12, that “groupoids can be thought of as ‘groups with many identities”; it is also useful to note that (loc.cit.) “the geometry underlying groupoids is that of based sets, i.e. sets with a chosen base point..” So, familiar things from, e.g., one’s first course in algebraic topology start appearing in one’s inner dialogue and the exotic quality of the whole business begins to wear down a little.

To drive this even further, and, indeed, to give a deeper rationale for the book under review, we note that the Theorem of Seifert-Van Kampen is featured explicitly: in Chapter 6 the 2-dimensional case is treated in connection with double groupoids, and then, in Chapter 8 (with Chapter 7 devoted to “The basics of crossed complexes”), the higher homotopy Seifert-Van Kampen Theorem is addressed.

So it is that we are dealing with some rather sophisticated and deep themes in algebraic topology, and the book under review gives an in-depth account of the according material, as can be done only by true insiders. Brown, Higgins, and Sivera are true proselytes for their cause, and take great pains to make it accessible to their audience: “The aim is for the major parts of this book to be readable by a graduate student acquainted with general topology, the fundamental group, notions of homotopy, and some basic methods of category theory. Many of these areas, including the concept of groupoid and its uses, are covered in Brown’s text, Topology and Groupoids … The only theory we have to assume for the Homotopy Classification Theorem … is some results on the geometric realization of cubical sets [which are ‘just covariant functors from the [box] category [of standard n-cubes] to the category of sets’].”

Thus, Nonabelian Algebraic Topology is a very ambitious pedagogical undertaking, aiming at presenting to a mathematically “young” audience some serious and often difficult mathematics: on. p.168, for example, we read that “[t]he homotopy classification of maps between topological spaces is among the most difficult areas in homotopy theory, and so this chapter [Ch. 11] is one of the most important in this book,” after which the authors proceed to discuss classifying spaces and characterize the aforementioned homotopy classification theorem (upcoming shortly) as a generalization of the Eilenberg-Mac Lane theorems for (whoa!) “a CW-complex … with skeletal filtration …” — pretty heavy stuff for the unsuspecting ...

But the book is very well crafted as an advanced text, so the committed beginning topologist should indeed be able to us it to great advantage. There is a very carefully presented xxxv-page Introduction, explaining not just how the book is structured and how its parts interrelate, but also, as we have already seen, gently introducing the reader to many of the main players, including some of the more exotic ones. (Also, the “Historical context diagram” on p.xix is a marvel worthy of Bourbaki after too much coffee and Gauloises — all kidding aside: it’s very useful, actually …) Then the book proper is split into three Parts the first two of which are roughly styled as, respectively, “develop[ing] that aspect of nonabelian algebraic topology related to the Seifert-Van Kampen Theorem in dimensions 1 and 2” (cf. p.3), and “obtain[ing] homotopical calculations using crossed complexes … [with] a Higher Homotopy Seifert- Van Kampen Theorem play[ing] a key role” (cf. p.207). After this, the book’s third Part (starting on p.439) is introduced with the passage: “In Part II we have explored the techniques of crossed complexes, and hope we have shown convincingly that they are a powerful tool in algebraic topology. In this part, we give the proofs of the main theorems on which those tools depend.” Then it’s on to “cubical ω-groupoids with connections.”

The book is very clearly written, heavily annotated and foot-noted, and sports examples as well as exercises woven through the text. All in all, it’s a very impressive piece of scholarship promising to have a beneficial pedagogical effect in topological circles.

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

Part I. 1- and 2-dimensional results

  • Introduction to Part I
  • History
  • Homotopy theory and crossed modules
  • Basic algebra of crossed modules
  • Coproducts of crossed $P$-modules
  • Induced crossed modules
  • Double groupoids and the 2-dimensional Seifert-van Kampen Theorem

Part II. Crossed complexes

  • Introduction to Part II
  • The basics of crossed complexes
  • The higher homotopy Seifert-van Kampen Theorem (HHSvKT) and its applications
  • Tensor products and homotopies of crossed complexes
  • Resolutions
  • The cubical classifying space of a crossed complex
  • Nonabelian cohomology: spaces, groupoids

Part III. Cubical $\omega$-groupoids

  • Introduction to Part III
  • The algebra of crossed complexes and cubical $\omega$-groupoids
  • The cubical homotopy $\omega$-groupoid of a filtered space
  • Tensor products and homotopies
  • Future directions?
  • Appendices
  • Bibliography
  • Glossary of symbols
  • Index