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Introduction to algebraic topology

Kammeyer, Holger.

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Introduction to algebraic topology

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Introduction to algebraic topology/ by Holger Kammeyer.
作者:	Kammeyer, Holger.
出版者:	Cham :Springer International Publishing : : 2022.,
面頁冊數:	viii, 182 p. :ill., digital ;24 cm.
內容註:	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.
Contained By:	Springer Nature eBook
標題:	Algebraic topology. -
電子資源:	https://doi.org/10.1007/978-3-030-98313-0
ISBN:	9783030983130

Introduction to algebraic topology
Kammeyer, Holger.

Introduction to algebraic topology[electronic resource] /by Holger Kammeyer. - Cham :Springer International Publishing :2022. - viii, 182 p. :ill., digital ;24 cm. - Compact textbooks in mathematics,2296-455X. - Compact textbooks in mathematics..

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.

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.

ISBN: 9783030983130

Standard No.: 10.1007/978-3-030-98313-0doiSubjects--Topical Terms:

532744
Algebraic topology.

LC Class. No.: QA612 / .K35 2022

Dewey Class. No.: 514.2

Introduction to algebraic topology
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520 $a 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.
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