1st Edition

Basic Algebraic Topology

ISBN 9781466562431
Published October 23, 2013 by Chapman and Hall/CRC
551 Pages 59 B/W Illustrations

USD $120.00

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Book Description

Building on rudimentary knowledge of real analysis, point-set topology, and basic algebra, Basic Algebraic Topology provides plenty of material for a two-semester course in algebraic topology.

The book first introduces the necessary fundamental concepts, such as relative homotopy, fibrations and cofibrations, category theory, cell complexes, and simplicial complexes. It then focuses on the fundamental group, covering spaces and elementary aspects of homology theory. It presents the central objects of study in topology visualization: manifolds. After developing the homology theory with coefficients, homology of the products, and cohomology algebra, the book returns to the study of manifolds, discussing Poincaré duality and the De Rham theorem. A brief introduction to cohomology of sheaves and Čech cohomology follows. The core of the text covers higher homotopy groups, Hurewicz’s isomorphism theorem, obstruction theory, Eilenberg-Mac Lane spaces, and Moore-Postnikov decomposition. The author then relates the homology of the total space of a fibration to that of the base and the fiber, with applications to characteristic classes and vector bundles. The book concludes with the basic theory of spectral sequences and several applications, including Serre’s seminal work on higher homotopy groups.

Thoroughly classroom-tested, this self-contained text takes students all the way to becoming algebraic topologists. Historical remarks throughout the text make the subject more meaningful to students. Also suitable for researchers, the book provides references for further reading, presents full proofs of all results, and includes numerous exercises of varying levels.

Table of Contents

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

Hints and Solutions



Exercises appear at the end of each chapter.

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Dr. Anant R. Shastri is a professor in the Department of Mathematics at the Indian Institute of Technology Bombay, where he has been teaching for over 20 years. His research focuses on the topology of matrix varieties.


"… 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, MAA Reviews, February 2014

"Similar to his other well-written textbook on differential topology, Professor Shastri’s book gives a detailed introduction to the vast subject of algebraic topology together with an abundance of carefully chosen exercises at the end of each chapter. The content of Professor Shastri’s book furnishes the necessary background to access many major achievements … [and] to explore current research works as well as possible applications to other branches of mathematics of modern algebraic topology."
—From the Foreword by Professor Peter Wong, Bates College, Lewiston, Maine, USA