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Publisher:

Chapman & Hall/CRC

Publication Date:

2013

Number of Pages:

535

Format:

Hardcover

Price:

89.95

ISBN:

9781466562431

Category:

Textbook

[Reviewed by , on ]

Michael Berg

02/27/2014

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

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.

**Introduction **

The Basic Problem

Fundamental Group

Function Spaces and Quotient Spaces

Relative Homotopy

Some Typical Constructions

Cofibrations

Fibrations

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

Applications

**Homology Groups**

Basic Homological Algebra

Singular Homology Groups

Construction of Some Other Homology Groups

Some Applications of Homology

Relation between π_{1} and *H*_{1}

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

**Cohomology **

Cochain Complexes

Universal Coefficient Theorem for Cohomology

Products in Cohomology

Some Computations

Cohomology Operations; Steenrod Squares

**Homology of Manifolds **

Orientability

Duality Theorems

Some Applications

de Rham Cohomology

**Cohomology of Sheaves **

Sheaves

Injective Sheaves and Resolutions

Cohomology of Sheaves

Čech Cohomology

**Homotopy Theory **

*H*-Spaces and *H*^{0}-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**

Warm-Up

Exact Couples

Algebra of Spectral Sequences

Leray–Serre Spectral Sequence

Some Immediate Applications

Transgression

Cohomology Spectral Sequences

Serre Classes

Homotopy Groups of Spheres

**Hints and Solutions**

**Bibliography **

**Index**

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