You are here

Basic Algebraic Topology and Its Applications

Mahima Ranjan Adhikari
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
, on

Given its chronological and evolutionary trajectory, algebraic topology is arguably the twentieth century’s most emblematic mathematical subject (well, perhaps algebraic geometry is a competitor for the title). After its spectacular inception with Poincaré’s Analysis Situs and its subsequent growth associated with Brouwer, Hopf, Alexandroff, Pontryagin, Hurewicz, and other such pioneers, we might identify the culmination of this first phase with the papers and books of Eilenberg-Steenrod and Eilenberg-Mac Lane: algebraic topology could now be characterized in terms of functors mapping from topological spaces to groups. Surely the emergence of this formalism qualifies as an exceptionally persuasive example of the power of abstraction and algebra, itself very young, at least in its axiomatic form: recall that Emmy Noether only died in 1935. Largely due to the labors of such scholars as H. Cartan, Serre, de Rham, (J. H. C.) Whitehead, and Adams, to name a few, such magical things as spectral sequences, ever more intricate homotopy theory, and sundry exotic flavors of cohomology entered the scene in the aftermath of World War II, and the subject has flourished in this mode ever since. It is without qualification a sine qua non for any serious mathematician today, particularly in light of the cross-pollination that occurs with everything from differential and algebraic geometry to physics.

Accordingly there are a host of fabulous books available, and to give a perfect number, here are six (although it’s not too difficult even to come up with twenty-eight, I guess), which, as is customary, I’ll just present by giving their authors: Eilenberg-Steenrod, Dold, Spanier, Maunder, Hatcher, and Bott-Tu. Yes, I know there are other favorites, but let’s not go down that path: isn’t it largely true that all these books are all homotopically equivalent?

By way of complete disclosure, the book I refer to the most myself (as a visitor to algebraic topology from number theory) is May’s A Concise Course in Algebraic Topology. My reasons for liking May’s book so much are many and varied, but one is his fine treatment of the subject of universal principal G-bundles for a topological group G, with close ties to simplicial homology: I needed to learn about the former’s construction for a paper I am working on, and this is a pretty esoteric business it seems. In any case, I had a hard time finding other satisfactory compact discussions of this theme (which, I admit, is more of a reflection on me than anything else).

Given that my head had been pretty much buried in May’s book, when I turned to another pursuit, namely going at this review, I figured that checking to see if Adhikari hits this sticky wicket were a good idea. And there it is on p. 219ff: an excellent discussion of these universal bundles as part of Chapter 5’s fifth section, titled “Homotopy properties of numerable principal G-bundles.” Adhikari actually follows the classical approach to this theme, with, indeed, his next section being devoted to the famous Milnor construction (in, of course, the homotopy category). I want to note explicitly that his treatment of what universality means in this context is particularly well done. This is actually something of a dicey business in that, at least to me who first saw it in a discussion (in a paper by Bott) explicitly focused on equivariant cohomology, there is something, if not unexpected, then at least unusual, about this definition. Adhikari not only makes it crystal clear, but presents a subsequent theorem that immediately elucidates and discloses the power and usefulness of this property.

It is also noteworthy that right after discussing fiber bundles in this fifth chapter, Adhikari goes on to the massively important topic of vector bundles, stressing right off the bat that “their homotopy classifications play a very important role in mathematics and physics.” He then launches into a very nice discussion of this material, including welcome categorical material (cf. his Corollary 5.7.14 on p. 227). Tangentially, online (on YouTube) there are very informative lectures available on geometry and physics done by Hirosi Ooguri of Caltech, in which the physicists’ angle on these beasties is particularly clearly presented (meaning that the mathematics is given much more of its due than might be the case).

Well, I guess it’s already clear that I am pretty enthusiastic about this book. It is rather comprehensive (at around 600 pages it would be difficult to avoid this), and it shows very good taste on the author’s part as far as what he’s chosen to do and how he’s chosen to do it. Homotopy comes first — well, second: the first chapter is devoted to prerequisites, and that really means a set of solid undergraduate courses in topology, yes, but also group theory, linear algebra (theorems and proofs, not just matrices and determinants), and such things as function spaces and manifolds; so I guess we’d want analysis and differential geometry for the very young.

After a beefy discussion of homotopy, Adhikari does fundamental groups and here it gets particularly cool: his Section 3.8 is devoted to the following beautiful applications that everyone should see, really: the Fundamental Theorem of Algebra, Brouwer’s Fixed Point Theorem (an alternate proof), Borsuk-Ulam, and Cauchy’s Integral Theorem. It’s a nice coverage of a spectrum, indicating the span and sweep of even this elementary part of algebraic topology.

Next: covering spaces, fiber and vector bundles, simplicial stuff, and then … more homotopy: higher homotopy groups, CW-complexes and homotopy (and it is truly a good idea to spend as much time on this theme as Adhikari does), and then products in homotopy. Evidently the latter chapter gets into some arcana, e.g. relations between Samelson and Whitehead products; he even goes after a theorem of J. Frank Adams and stuff about the 7-sphere (again, Milnor is smiling).

A little over halfway through, at around p. 350, we finally get to (co)homology, but not only is it well worth waiting for, but, as we’ve just seen, what has come before is fabulous and very important. It’s just that as a fellow traveler from Zahlentheorie, my own history has been dominated by cohomology (it all started with class field theory — danke sehr, Emil Artin and John Tate), and I’m something of a late comer to the inner life of algebraic topology. That said, Adhikari’s coverage of (co)homology is splendid: after a very extensive discussion of homology and cohomology theories in Chapter 10, capped off with three sexy applications (the Jordan curve theorem, a bit on homology groups of spheres, and invariance of domain), Eilenberg and Steenrod take the stage for three chapters. After this we hit the stretch with a chapter on applications of the foregoing, spectral (co)homology, and obstruction theory. Adhikari closes his opus with material having to do with “more relations between homology and homotopy” (Hurewicz and more!), and as icing on the (big) cake, a brief history of the whole subject of algebraic topology.

Wow! What a nice book. I’m glad I have a copy.

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

See the table of contents in the publisher's webpage.