Abstract
This is the core chapter of the book. The classification theorem for compact surfaces with or without boundaries is stated and proved. The key is to define the notion of a cell complex. Every cell complex can be refined to a triangulation. Then, a combinatorial argument is used to show that every cell complex can be converted to a canonical form involving a regular polygon. Homology is used to show that distinct normal forms are not homeomorphic. We also show how the normal form of a surface yields a presentation of its fundamental group. Other proofs are discussed, in particular, Conway’s ZIP proof.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
L.V. Ahlfors, L. Sario, Riemann Surfaces, Princeton Math. Series, No. 2 (Princeton University Press, Princeton, 1960)
M.A. Amstrong, Basic Topology, UTM, 1st edn. (Springer, New York, 1983)
P. Andrews, The classification of surfaces. Am. Math. Mon. 95(9), 861–867 (1988)
E.D. Bloch, A First Course in Geometric Topology and Differential Geometry, 1st edn. (Birkhäuser, Boston, 1997)
C.E. Burgess, Classification of surfaces. Am. Math. Mon. 92(5), 349–354 (1985)
P.H. Doyle, D.A. Moran, A short proof that compact 2-manifolds can be triangulated. Invent. Math. 5(2), 160–162 (1968)
M. Fréchet, K. Fan, Invitation to Combinatorial Topology, 1st edn. (Dover, New York, 2003)
G.K. Francis, J.R. Weeks, Conway’s ZIP proof. Am. Math. Mon. 106(5), 393–399 (1999)
W. Fulton, Algebraic Topology, A first course, GTM, No. 153, 1st edn. (Springer, New York, 1995)
M. Henle, A Combinatorial Introduction to Topology, 1st edn. (Dover, New York, 1994)
D. Hilbert, S. Cohn–Vossen, Geometry and the Imagination (Chelsea, New York, 1952)
B.M. Kerékjártó, Vorlesungen über Topologie, I, 1st edn. (Julius Springer, Berlin, 1923)
L.C. Kinsey, Topology of Surfaces, UTM, 1st edn. (Springer, New York, 1993)
J.M. Lee, Introduction to Topological Manifolds, GTM vol. 202, 2nd edn. (Springer, New York, 2011)
F. Levi, Geometrische Konfigurationen, 1st edn. (Leipzig, Hirzel, 1929)
W.S. Massey, Algebraic Topology: An Introduction, GTM No. 56, 2nd edn. (Springer, New York, 1987)
J.R. Munkres, Topology, 2nd edn. (Prentice Hall, Englewood Cliffs, 2000)
K. Reidemeister, Einführung in die Kombinatorische Topologie, 1st edn. (Braunschweig, Vieweg, 1932)
H. Seifert, W. Threlfall, A Textbook of Topology, 1st edn. (Academic, New York, 1980)
C. Thomassen, The Jordan-Schonflies Theorem and the classification of surfaces. Am. Math. Mon. 99(2), 116–131 (1992)
C. Thomassen, B. Mohar, Graphs on Surfaces, 1st edn. (The Johns Hopkins University Press, London, 2001)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Gallier, J., Xu, D. (2013). The Classification Theorem for Compact Surfaces. In: A Guide to the Classification Theorem for Compact Surfaces. Geometry and Computing, vol 9. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34364-3_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-34364-3_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34363-6
Online ISBN: 978-3-642-34364-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)