Introduction to Algebraic Topology

This textbook provides a modern categorical approach to algebraic topology,with an outline of category theory,van Kampen's theorem,cofibrations,homotopy pushouts,simplicial homology,the Eilenberg-Steenrod axioms,the simplicial approximation theorem,the Mayer-Vietoris sequence,and cellular homology. It is suitable for a single-semester graduate course or self-study and includes numerous examples,exercises,and motivating remarks.

Format: Paperback / softback
Length: 182 pages
Publication date: 21 June 2022
Publisher: Springer Nature Switzerland AG

This textbook offers a concise and comprehensive introduction to algebraic topology,following a modern categorical approach from the very beginning. It provides ample motivation throughout,making it an ideal first encounter with the field for students. Topics are treated in a self-contained manner,making it a convenient resource for instructors seeking a comprehensive overview of the area.

The book 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.

This textbook 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.

In conclusion,Introduction to Algebraic Topology is a valuable resource for students and instructors alike,providing a concise and comprehensive introduction to the field. Its modern categorical approach,self-contained treatment of topics,and ample motivation make it an ideal first encounter with algebraic topology.

Weight: 334g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030983123
Edition number: 1st ed. 2022