551 Pages 59 B/W Illustrations
    by Chapman & Hall

    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.

    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.


    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