Introduction to Algebraic Topology

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.
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.

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

Autor*in

Holger Kammeyer

Themen in »Introduction to Algebraic Topology«

uniqueness of ordinary homology 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

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“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)

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