By Jean H Gallier; Dianna Xu

ISBN-10: 3642343643

ISBN-13: 9783642343643

This welcome boon for college kids of algebraic topology cuts a much-needed crucial course among different texts whose therapy of the class theorem for compact surfaces is both too formalized and complicated for these with no precise history wisdom, or too casual to come up with the money for scholars a entire perception into the topic. Its devoted, student-centred technique info a near-complete evidence of this theorem, generally prominent for its efficacy and formal good looks. The authors current the technical instruments had to installation the strategy successfully in addition to demonstrating their use in a truly dependent, labored instance. learn more... The class Theorem: casual Presentation -- Surfaces -- Simplices, Complexes, and Triangulations -- the elemental crew, Orientability -- Homology teams -- The category Theorem for Compact Surfaces. The type Theorem: casual Presentation -- Surfaces -- Simplices, Complexes, and Triangulations -- the elemental staff -- Homology teams -- The class Theorem for Compact Surfaces

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W. von Dyck, Beitr¨age zur analysis situs. Mathematische Annalen 32, 457–512 (1888) 5. M. Fr´echet, K. 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.

Let us choose some point a in E (a base point), and consider all closed paths W Œ0; 1 ! 1/ D a. We can compose closed paths 1 ; 2 based at a, and consider the inverse 1 of a closed path, but unfortunately, the operation of composition of 1 closed paths is not associative, and is not the identity in general. In order to obtain a group structure, we define a notion of equivalence of closed paths under continuous deformations. Actually, such a notion can be defined for any two paths with the same origin and extremity, and even for continuous maps.

In this case, we say that 1 and homotopic) and this is denoted by 1 2. 1 The Fundamental Group 39 γ1 Fig. 2 A path homotopy between 1 and 2 a b γ2 Given any two continuous maps f1 W X ! Y and f2 W X ! Y between two topological spaces X and Y , a map F W X Œ0; 1 ! t/; for all t 2 X . We say that f1 and f2 are homotopic, and this is denoted by f1 ' f2 . t; u/ from a to b, giving a deformation of the path 1 into the path 2 , and leaving the endpoints a and b fixed, as illustrated in Fig. 2. Similarly, a homotopy between two continuous maps f1 and f2 is a continuous family of maps ft giving a deformation of f1 into f2 .

### A guide to the classification theorem for compact surfaces by Jean H Gallier; Dianna Xu

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