A guide to the classification theorem for compact surfaces / Jean Gallier, Dianna Xu.
- Heidelberg ; New York : Springer, c2013.
- Geometry and computing ; 9.
Geometry and computing, 1866-6795 ; v. 9
xii, 178 p. : ill. (some col.) ; 24 cm.
- Topological algebras.
- 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-centered 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.
- The classification theorem: informal presentation
Simplices, complexes, and triangulations
The fundamental group, orientability
The classification theorem for compact surfaces
Viewing the real projective plane in R³; the cross-cap and the Steiner roman surface
Proof of proposition 5.1
History of the classification theorem
Every surface can be triangulated.
- Includes bibliographical references and indexes.
- Xu, Dianna.
- Publisher Number:
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