Geometry and Computing

A Guide to the Classification Theorem for Compact Surfaces

Authors: Gallier, Jean, Xu, Dianna

  • Student-centred guide offering comprehensive—and comprehensible—treatment of the classification theorem for compact surfaces
  • A short proof using graph theory (due to Thomassen) that every compact surface can be triangulated
  • Accessible to undergraduate students without assuming too much background ​
see more benefits

Buy this book

eBook $34.99
price for USA (gross)
  • ISBN 978-3-642-34364-3
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $49.95
price for USA
  • ISBN 978-3-642-34363-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
About this Textbook

This welcome boon for students of algebraic topology cuts a much-needed central path between other texts whose treatment of the classification theorem for compact surfaces is either too formalized and complex for those without detailed background knowledge, or too informal to afford students a comprehensive insight into the subject. Its dedicated, student-centred approach details a near-complete proof of this theorem, widely admired for its efficacy and formal beauty. The authors present the technical tools needed to deploy the method effectively as well as demonstrating their use in a clearly structured, worked example.

Ideal for students whose mastery of algebraic topology may be a work-in-progress, the text introduces key notions such as fundamental groups, homology groups, and the Euler-Poincaré characteristic. These prerequisites are the subject of detailed appendices that enable focused, discrete learning where it is required, without interrupting the carefully planned structure of the core exposition. Gently guiding readers through the principles, theory, and applications of the classification theorem, the authors aim to foster genuine confidence in its use and in so doing encourage readers to move on to a deeper exploration of the versatile and valuable techniques available in algebraic topology.

Reviews

From the reviews:

“The book would make an excellent graduate or advanced undergraduate project. … audience would be instructors or researchers looking to reconnect to some topology fundamentals. Practitioners in topology-adjacent fields might find it valuable as a concise reference. … a textbook or supplemental reference for a second topology course with some careful planning … . Academic libraries should strongly consider this book as it offers a broad look at material common to a standard undergraduate mathematics curriculum.” (Bill Wood, MAA Reviews, March, 2014)

“This book aims to give students … access to a true classic in algebraic topology: the classification of compact surfaces (up to homeomorphism). … The book includes many historical comments throughout and contains lots of pictures­­––both of the mathematicians involved in the classification theorem and of examples illustrating the mathematical content. Many examples are given that help one understand the basic tools … .” (Clara Lӧh, Mathematical Reviews, December, 2013)

“This highly focused book by Gallier (Univ. of Pennsylvania) and Xu (Bryn Mawr College) does both, delivering rigor to undergraduates by developing minimal doses of homotopy and homology theory, and without even presuming familiarity with group theory. … The present careful treatment of a major result that draws from several branches of mathematics makes the book an excellent resource for a capstone course. Summing Up: Highly recommended. Upper-division undergraduates and above.” (D. V. Feldman, Choice, Vol. 51 (1), September, 2013)

“The book is geared toward an audience including first-year graduate students but also strongly motivated upper-level undergraduates … . the book under review offers a beautiful introduction to one of the oldest and most fascinating theorems in algebraic topology … . the historical remarks concerning the classification theorem for compact surfaces are highly enlightening to anyone interested in topology, and the many related biographies and photographs are just as entertaining. … No doubt, this is an introductory text of remarkable didactic value.” (Werner Kleinert, zbMATH, Vol. 1270, 2013)


Table of contents (6 chapters)

  • The Classification Theorem: Informal Presentation

    Gallier, Jean (et al.)

    Pages 1-20

  • Surfaces

    Gallier, Jean (et al.)

    Pages 21-25

  • Simplices, Complexes, and Triangulations

    Gallier, Jean (et al.)

    Pages 27-35

  • The Fundamental Group, Orientability

    Gallier, Jean (et al.)

    Pages 37-51

  • Homology Groups

    Gallier, Jean (et al.)

    Pages 53-77

Buy this book

eBook $34.99
price for USA (gross)
  • ISBN 978-3-642-34364-3
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $49.95
price for USA
  • ISBN 978-3-642-34363-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
Loading...

Recommended for you

Loading...

Bibliographic Information

Bibliographic Information
Book Title
A Guide to the Classification Theorem for Compact Surfaces
Authors
Series Title
Geometry and Computing
Series Volume
9
Copyright
2013
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-34364-3
DOI
10.1007/978-3-642-34364-3
Hardcover ISBN
978-3-642-34363-6
Series ISSN
1866-6795
Edition Number
1
Number of Pages
XII, 178
Number of Illustrations and Tables
58 b/w illustrations, 20 illustrations in colour
Topics