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Basic Algebraic Topology

Anant R. Shastri
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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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.

The Basic Problem
Fundamental Group
Function Spaces and Quotient Spaces
Relative Homotopy
Some Typical Constructions
Categories and Functors

Cell Complexes and Simplicial Complexes
Basics of Convex Polytopes
Cell Complexes
Product of Cell Complexes
Homotopical Aspects
Cellular Maps
Abstract Simplicial Complexes
Geometric Realization of Simplicial Complexes
Barycentric Subdivision
Simplicial Approximation
Links and Stars

Covering Spaces and Fundamental Group
Basic Definitions
Lifting Properties
Relation with the Fundamental Group
Classification of Covering Projections
Group Action
Pushouts and Free Products
Seifert–van Kampen Theorem

Homology Groups
Basic Homological Algebra
Singular Homology Groups
Construction of Some Other Homology Groups
Some Applications of Homology
Relation between π1 and H1
All Postponed Proofs

Topology of Manifolds
Set Topological Aspects
Triangulation of Manifolds
Classification of Surfaces
Basics of Vector Bundles

Universal Coefficient Theorem for Homology
Method of Acyclic Models
Homology with Coefficients: The Tor Functor
Kűnneth Formula

Cochain Complexes
Universal Coefficient Theorem for Cohomology
Products in Cohomology
Some Computations
Cohomology Operations; Steenrod Squares

Homology of Manifolds
Duality Theorems
Some Applications
de Rham Cohomology

Cohomology of Sheaves
Injective Sheaves and Resolutions
Cohomology of Sheaves
Čech Cohomology

Homotopy Theory
H-Spaces and H0-Spaces
Higher Homotopy Groups
Change of Base Point
The Hurewicz Isomorphism
Obstruction Theory
Homotopy Extension and Classification
Eilenberg–Mac Lane Spaces
Moore–Postnikov Decomposition
Computation with Lie Groups and Their Quotients
Homology with Local Coefficients

Homology of Fibre Spaces
Generalities about Fibrations
Thom Isomorphism Theorem
Fibrations over Suspensions
Cohomology of Classical Groups

Characteristic Classes
Orientation and Euler Class
Construction of Steifel–Whitney Classes and Chern Classes
Fundamental Properties
Splitting Principle and Uniqueness
Complex Bundles and Pontrjagin Classes

Spectral Sequences
Exact Couples
Algebra of Spectral Sequences
Leray–Serre Spectral Sequence
Some Immediate Applications
Cohomology Spectral Sequences
Serre Classes
Homotopy Groups of Spheres

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