1st Edition

An Illustrated Introduction to Topology and Homotopy

By Sasho Kalajdzievski Copyright 2015
    488 Pages 460 B/W Illustrations
    by Chapman & Hall

    An Illustrated Introduction to Topology and Homotopy explores the beauty of topology and homotopy theory in a direct and engaging manner while illustrating the power of the theory through many, often surprising, applications. This self-contained book takes a visual and rigorous approach that incorporates both extensive illustrations and full proofs.

    The first part of the text covers basic topology, ranging from metric spaces and the axioms of topology through subspaces, product spaces, connectedness, compactness, and separation axioms to Urysohn’s lemma, Tietze’s theorems, and Stone-Čech compactification. Focusing on homotopy, the second part starts with the notions of ambient isotopy, homotopy, and the fundamental group. The book then covers basic combinatorial group theory, the Seifert-van Kampen theorem, knots, and low-dimensional manifolds. The last three chapters discuss the theory of covering spaces, the Borsuk-Ulam theorem, and applications in group theory, including various subgroup theorems.

    Requiring only some familiarity with group theory, the text includes a large number of figures as well as various examples that show how the theory can be applied. Each section starts with brief historical notes that trace the growth of the subject and ends with a set of exercises.

    Sets, Numbers, Cardinals, and Ordinals
    Sets and Numbers
    Sets and Cardinal Numbers
    Axiom of Choice and Equivalent Statements

    Metric Spaces: Definition, Examples, and Basics
    Metric Spaces: Definition and Examples
    Metric Spaces: Basics

    Topological Spaces: Definition and Examples
    The Definition and Some Simple Examples
    Some Basic Notions
    Dense and Nowhere Dense Sets
    Continuous Mappings

    Subspaces, Quotient Spaces, Manifolds, and CW-Complexes
    Quotient Spaces
    The Gluing Lemma, Topological Sums, and Some Special Quotient Spaces
    Manifolds and CW-Complexes

    Products of Spaces
    Finite Products of Spaces
    Infinite Products of Spaces
    Box Topology

    Connected Spaces and Path Connected Spaces
    Connected Spaces: Definition and Basic Facts
    Properties of Connected Spaces
    Path Connected Spaces
    Path Connected Spaces: More Properties and Related Matters
    Locally Connected and Locally Path Connected Spaces

    Compactness and Related Matters
    Compact Spaces: Definition
    Properties of Compact Spaces
    Compact, Lindelöf, and Countably Compact Spaces
    Bolzano, Weierstrass, and Lebesgue
    Infinite Products of Spaces and Tychonoff Theorem

    Separation Properties
    The Hierarchy of Separation Properties
    Regular Spaces and Normal Spaces
    Normal Spaces and Subspaces

    Urysohn, Tietze, and Stone-Čech
    Urysohn Lemma
    The Tietze Extension Theorem
    Stone-Čech Compactification

    Isotopy and Homotopy
    Isotopy and Ambient Isotopy
    Homotopy and Paths
    The Fundamental Group of a Space

    The Fundamental Group of a Circle and Applications
    The Fundamental Group of a Circle
    Brouwer Fixed Point Theorem and the Fundamental Theorem of Algebra
    The Jordan Curve Theorem

    Combinatorial Group Theory
    Group Presentations
    Free Groups, Tietze, Dehn
    Free Products and Free Products with Amalgamation

    Seifert–van Kampen Theorem and Applications
    Seifert–van Kampen Theorem
    Seifert–van Kampen Theorem: Examples
    The Seifert–van Kampen Theorem and Knots
    Torus Knots and Alexander’s Horned Sphere

    On Classifying Manifolds and Related Topics
    Compact 2-Manifolds: Preliminary Results
    Compact 2-Manifolds: Classification
    Regarding Classification of CW-Complexes and Higher Dimensional Manifolds
    Higher Homotopy Groups: A Brief Overview

    Covering Spaces, Part 1
    Covering Spaces: Definition, Examples, and Preliminaries
    Lifts of Paths
    Lifts of Mappings
    Covering Spaces and Homotopy

    Covering Spaces, Part 2
    Covering Spaces and Sheets
    Covering Trans formations
    Covering Spaces and Groups Acting Properly Discontinuously
    Covering Spaces: Existence
    The Borsuk–Ulam Theorem

    Applications in Group Theory
    Cayley Graphs and Covering Spaces
    Topographs and Presentations
    Subgroups of Free Groups
    Two Subgroup Theorems



    Sasho Kalajdzievski

    "… reflects interesting aspects and will find its readers."
    —Zentralblatt MATH
    , 1323

    "… an ideal college or university textbook and an invaluable addition to academic library mathematical studies reference collections."
    Library Bookwatch, May 2015