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Introduction to Algebraic Topology

Birkhäuser

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  • 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

Part of the book series: Compact Textbooks in Mathematics (CTM)

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  • ISBN: 978-3-030-98313-0
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Table of contents (6 chapters)

  1. Front Matter

    Pages i-viii
  2. Basic Notions of Category Theory

    • Holger Kammeyer
    Pages 1-31
  3. Homology: Ideas and Axioms

    • Holger Kammeyer
    Pages 59-78
  4. Singular Homology

    • Holger Kammeyer
    Pages 79-101
  5. Homology: Computations and Applications

    • Holger Kammeyer
    Pages 103-133
  6. Cellular Homology

    • Holger Kammeyer
    Pages 135-167
  7. Back Matter

    Pages 169-182

About this book

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.

Keywords

  • uniqueness of ordinary homology
  • Algebraic topology
  • 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

Authors and Affiliations

  • Mathematical Institute, Heinrich Heine University Düsseldorf, Düsseldorf, Germany

    Holger Kammeyer

About the author

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 ℓ²-invariants on which he has authored the textbook Introduction to ℓ²-invariants (Lecture Notes in Mathematics, Volume 2247, Springer).

Bibliographic Information

Buying options

eBook
USD 39.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-98313-0
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 54.99
Price excludes VAT (USA)