Introduction to algebraic topology

by Kammeyer, Holger

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Title Statements

Full title :Introduction to algebraic topology / Holger Kammeyer.

Summary

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. 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.

Edition Statements

Publication Date:©2022

Publication Statements

Publisher(s):Birkhäuser
Place of Publication:Cham
Language:eng
Publication Date:©2022

The Physical Object

Description:1 online resource : : illustrations.

Edition Identifiers

ISBN:9783030983130, 3030983137
DOI:10.1007/978-3-030-98313-0
oclc:on1331707569
LC Classification:QA612
Dewey Decimal:514/.2

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.