This textbook provides a succinct introduction to algebraic topology. It follows a modern categorical approach from the beginning and gives ample motivation throughout so that students will find this an ideal first encounter to the field. Topics are treated in a self-contained manner, making this a convenient resource for instructors searching for a comprehensive overview of the area.
It begins with an outline of category theory, establishing the concepts of functors, natural transformations, adjunction, limits, and colimits. As a first application, van Kampen's theorem is proven in the groupoid version. Following this, an excursion to cofibrations and homotopy pushouts yields an alternative formulation of the theorem that puts the computation of fundamental groups of attaching spaces on firm ground. Simplicial homology is then defined, motivating the Eilenberg-Steenrod axioms, and the simplicial approximation theorem is proven. After verifying the axioms for singular homology, various versions of the Mayer-Vietoris sequence are derived and it is shown that homotopy classes of self-maps of spheres are classified by degree.The final chapter discusses cellular homology of CW complexes, culminating in the uniqueness theorem for ordinary homology.
Author(s): Holger Kammeyer
Series: Compact Textbooks in Mathematics
Edition: 1
Publisher: Birkhäuser
Year: 2022
Language: English
Pages: 182
City: Cham, Switzerland
Tags: Algebraic Topology, Categories, Fundamental Groupoid, Homology, CW Complexes
Preface
Contents
1 Basic Notions of Category Theory
1.1 Categories
1.2 Functors
1.3 Natural Transformations
1.4 Adjunction
1.5 Limits and Colimits
Exercises
2 Fundamental Groupoid and van Kampen's Theorem
2.1 The Fundamental Groupoid
2.2 Van Kampen's Theorem
2.3 Cofibrations and Homotopy Pushouts
2.4 Computing Fundamental Groups
2.5 Higher Homotopy Groups
Exercises
3 Homology: Ideas and Axioms
3.1 The Idea of Homology
3.2 Simplicial Homology
3.3 Relative Simplicial Homology with Coefficients
3.4 The Eilenberg–Steenrod Axioms for Homology
3.5 Simplicial Approximation
Exercises
4 Singular Homology
4.1 The Definition of Singular Homology
4.2 The Long Exact Sequence of a Pair of Spaces
4.3 Homotopy Invariance
4.4 Excision
4.5 Singular Homology in Degree Zero and One
Exercises
5 Homology: Computations and Applications
5.1 Relative vs. Absolute Homology
5.2 Simplicial and Singular Homology Agree
5.3 The Mayer–Vietoris Sequence
5.4 Degree
5.5 Applications
The Fundamental Theorem of Algebra
Invariance of Dimension
Nonexistence of Retractions
The Brouwer Fixed Point Theorem
The Borsuk–Ulam Theorem
The Ham Sandwich Theorem
Exercises
6 Cellular Homology
6.1 CW Complexes
6.2 Cellular Homology and Euler Characteristic
6.3 Computing Cellular Homology
6.4 Uniqueness of Ordinary Homology
6.5 How to Proceed
Exercises
A Quotient Topology
Bibliography
List of Notation
List of Notation
Index
