Basic Algebraic Topology

The book under review is a somewhat compact (if you’ll pardon the expression) and fast-paced treatment of algebraic topology for an audience of beginning graduate students. The book is at once purposely narrow and ambitious in its scope — perhaps this juggling of apparently opposite objectives cannot be avoided when it comes to presenting newcomers with the huge and sophisticated tool-kit of this major branch of modern mathematics.

Shastri characterizes algebraic topology as a set of answers, so to speak, to the basic question “when are two topological spaces homeomorphic?”, a fundamental theme which has indeed given rise to a panoply of topological invariants and marvelous constructions that transcend the boundaries of topology. These have led to such sweeping “philosophies” as the method of categories and functors, again spilling over into mathematics at large. The according impact of algebraic topology in the broadest sense is really best conveyed in Jean Dieudonné’s titanic History of Algebraic and Differential Topology (1900–1960).

But what of Shastri’s book? Well, it’s all there, in essentially historical order. After developing the basics, including the fundamental group, relative homotopy, cofibrations and fibrations and a bit of category theory, the march is on: simplicial complexes, covering spaces, homology (especially singular homology), manifolds and vector bundles, the universal coefficient theorem, cohomology, and then a focus on manifolds and de Rham cohomology. It’s certainly something of a crescendo, and I should say that, in rough terms, this collectively makes for a good first semester curriculum. Thereafter Shastri hits sheaf cohomology (Serre’s Faisceaux Algébriques Cohérents, not Grothendieck’s SGA) and homotopy theory (including higher homotopy groups, obstruction theory, Eilenberg-Mac Lane spaces, the Moore-Postnikov decomposition, and homology with local coefficients). He goes on to finish with three beefy themes: the homology of fiber spaces (including the Thom isomorphism theorem), characteristic classes, and spectral sequences. The very last section of the book addresses nothing less than Serre’s famous result on the homotopy groups of spheres: this culminating discussion is only two pages long, but requires a huge proportion of the material that precedes it. This condition in a sense characterizes the entire book — it’s an awful lot of wonderful material covered rather tersely. After all the book is only around 500 pages long, and for this much topology that’s not all that much.

So does Shastri pull it off? Does he succeed in presenting a viable text for a year’s course in algebraic topology covering such a wealth of material? I think he does. Indeed, treated right, Basic Algebraic Topology will serve well as both a successful class-room tool and a source for serious self-study. But be forewarned: Shastri wastes no time, and there is a lot of ground to cover

All in all, I think Basic Algebraic Topology is a good graduate text: the book is well-written and there are many well-chosen examples and a decent number of exercises. It meets its ambitious goals and should succeed in leading a lot of solid graduate students, as well as working mathematicians from other specialties seeking to learn this subject, deeper and deeper into its workings and subtleties.

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