Overview
- Offers a modern and concise introduction to algebraic topology
- Features a short chapter on category theory, ideal for a first encounter to the field
- Provides motivation for graduate students studying in the classroom
Part of the book series: Compact Textbooks in Mathematics (CTM)
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About this book
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 axiomsfor 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.
Introduction to Algebraic Topology is suitable for a single-semester graduate course on algebraic topology. It can also be used for self-study, with numerous examples, exercises, and motivating remarks included.
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 axiomsfor 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.
Introduction to Algebraic Topology is suitable for a single-semester graduate course on algebraic topology. It can also be used for self-study, with numerous examples, exercises, and motivating remarks included.
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Keywords
- uniqueness of ordinary homology
- Algebraic topology
- Category theory
- Singular homology
- Singular homology vs simplicial
- Cellular homology
- Fundamental groupoid
- Fundamental groupoid category
- Fundamental groupoid van Kampen
- Homotopy pushouts
- Homotopy category
- Homotopy pushout category
- Simplicial homology
- Relative homology absolute
- Absolute homology relative
Table of contents (7 chapters)
Reviews
“Throughout the book there are many diagrams, pictures and nice examples to explain the concepts. … The reviewer finds reading the book a pleasure. … The reviewer recommends the book to any student taking a first course in algebraic topology … .” (Man-Ho Ho, Mathematical Reviews, April, 2023)
“The book is beautifully laid out, with clearly marked offset examples and remarks, and some biographical information about some of the mathematicians referenced. The number of exercises is modest, with a few at the end of each chapter, but they seem to be thoughtfully designed to introduce new key concepts. This book should be a valuable resource for anyone teaching a course at this level.” (Julie Bergner, zbMATH 1502.55001, 2023)
Authors and Affiliations
About the author
Holger Kammeyer is Assistant Professor of Algebra and Geometry at the University of Düsseldorf. His research interests include algebraic topology as well as arithmetic and profinite groups. A particular field of his expertise is the theory of ℓ²-invariants on which he has authored the textbook Introduction to ℓ²-invariants (Lecture Notes in Mathematics, Volume 2247, Springer).
Bibliographic Information
Book Title: Introduction to Algebraic Topology
Authors: Holger Kammeyer
Series Title: Compact Textbooks in Mathematics
DOI: https://doi.org/10.1007/978-3-030-98313-0
Publisher: Birkhäuser Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
Softcover ISBN: 978-3-030-98312-3Published: 21 June 2022
eBook ISBN: 978-3-030-98313-0Published: 20 June 2022
Series ISSN: 2296-4568
Series E-ISSN: 2296-455X
Edition Number: 1
Number of Pages: VIII, 182
Number of Illustrations: 10 b/w illustrations
Topics: Algebraic Topology, Category Theory, Homological Algebra