Basic Algebraic Topology: 1st Edition (Hardback) book cover

Basic Algebraic Topology

1st Edition

By Anant R. Shastri

Chapman and Hall/CRC

551 pages | 59 B/W Illus.

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pub: 2013-10-23
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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.

Reviews

"… 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

Table of Contents

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 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

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 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

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

Exercises appear at the end of each chapter.

About the Author

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.

Subject Categories

BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT002000
MATHEMATICS / Algebra / General
MAT012000
MATHEMATICS / Geometry / General