An Illustrated Introduction to Topology and Homotopy

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-Cech 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.

TOPOLOGYSets, Numbers, Cardinals, and Ordinals Sets and Numbers Sets and Cardinal Numbers Axiom of Choice and Equivalent Statements Metric Spaces: Definition, Examples, and BasicsMetric Spaces: Definition and Examples Metric Spaces: Basics Topological Spaces: Definition and ExamplesThe 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 SpacesFinite 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 CompactificationInfinite Products of Spaces and Tychonoff Theorem Separation PropertiesThe Hierarchy of Separation PropertiesRegular Spaces and Normal SpacesNormal Spaces and Subspaces Urysohn, Tietze, and Stone-Cech Urysohn Lemma The Tietze Extension Theorem Stone-Cech Compactification HOMOTOPYIsotopy 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 TheoryGroup PresentationsFree 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 Links 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 ManifoldsHigher 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 formationsCovering 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