Introduction to Algebraic Topology

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Sprache: Englisch

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Beschreibung

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.

Über den Autor

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 ¿²-invariants on which he has authored the textbook Introduction to ¿²-invariants (Lecture Notes in Mathematics, Volume 2247, Springer).

Zusammenfassung

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

Inhaltsverzeichnis

Basic notions of category theory.- Fundamental groupoid and van Kampen's theorem.- Homology: ideas and axioms.- Singular homology.- Homology: computations and applications.- Cellular homology.- Appendix: Quotient topology.

Details

Erscheinungsjahr:	2022
Fachbereich:	Geometrie
Genre:	Mathematik
Rubrik:	Naturwissenschaften & Technik
Medium:	Taschenbuch
Reihe:	Compact Textbooks in Mathematics
Inhalt:	viii 182 S. 10 s/w Illustr. 182 p. 10 illus.
ISBN-13:	9783030983123
ISBN-10:	3030983129
Sprache:	Englisch
Ausstattung / Beilage:	Paperback
Einband:	Kartoniert / Broschiert
Autor:	Kammeyer, Holger
Auflage:	1st ed. 2022
Hersteller:	Springer International Publishing Springer International Publishing AG Compact Textbooks in Mathematics
Maße:	235 x 155 x 11 mm
Von/Mit:	Holger Kammeyer
Erscheinungsdatum:	21.06.2022
Gewicht:	0,335 kg

Artikel-ID: 121187873