By Jean H Gallier; Dianna Xu

This welcome boon for college kids of algebraic topology cuts a much-needed primary direction among different texts whose therapy of the type theorem for compact surfaces is both too formalized and intricate for these with out particular heritage wisdom, or too casual to find the money for scholars a complete perception into the topic. Its committed, student-centred procedure info a near-complete evidence of this theorem, greatly sought after for its efficacy and formal good looks. The authors current the technical instruments had to install the strategy successfully in addition to demonstrating their use in a basically dependent, labored instance. learn more... The class Theorem: casual Presentation -- Surfaces -- Simplices, Complexes, and Triangulations -- the elemental workforce, Orientability -- Homology teams -- The class Theorem for Compact Surfaces. The category Theorem: casual Presentation -- Surfaces -- Simplices, Complexes, and Triangulations -- the elemental crew -- Homology teams -- The type Theorem for Compact Surfaces

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Fan, Invitation to Combinatorial Topology, 1st edn. (Dover, New York, 2003) 6. D. Hilbert, S. Cohn–Vossen, Anschauliche Geometrie, 2nd edn. (Springer, New York, 1996) 7. D. Hilbert, S Cohn–Vossen, Geometry and the Imagination (Chelsea, New York, 1952) 8. C. Jordan, Sur la d´eformation des surfaces. J. de Math´ematiques Pures et Appliqu´ees 2e s´erie 11, 105–109 (1866) ¨ 9. F. Klein, Uber Riemanns Theorie der Algebraischen Funktionen und Ihrer Integrale, 1st edn. G. Teubner, Leipzig, 1882) 10. B.

Given any sequence of n points a1 ; : : : ; an in an affine space E , an affine combination of these points is a linear combination 1 a1 with i C C n an ; 2 R, and with the restriction that 1 C C n D 1: J. Gallier and D. 1007/978-3-642-34364-3 3, © Springer-Verlag Berlin Heidelberg 2013 ( ) 27 28 3 Simplices, Complexes, and Triangulations Condition ( ) ensures that an affine combination does not depend on the choice of an origin. An affine combination is a convex combination if the scalars i satisfy the extra conditions i 0, in addition to 1 C C n D 1.

Rotman, Introduction to Algebraic Topology, GTM No. 119, 1st edn. 1 The Fundamental Group If we want to somehow classify surfaces, we have to deal with the issue of deciding when we consider two surfaces to be equivalent. It seems reasonable to treat homeomorphic surfaces as equivalent, but this leads to the problem of deciding when two surfaces are not homeomorphic, which is a very difficult problem. One way to approach this problem is to forget some of the topological structure of a surface and look for more algebraic objects that can be associated with a surface.