1st Edition

# An Illustrated Introduction to Topology and Homotopy

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

485 Pages
by Chapman & Hall

Also available as eBook on:

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.

TOPOLOGY
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
Bases
Dense and Nowhere Dense Sets
Continuous Mappings

Subspaces, Quotient Spaces, Manifolds, and CW-Complexes
Subspaces
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
Compactification
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

HOMOTOPY
Isotopy and Homotopy
Isotopy and Ambient Isotopy
Homotopy
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
1-Manifolds
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

Bibliography

### Biography

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