By Jean H Gallier; Dianna Xu

This welcome boon for college students of algebraic topology cuts a much-needed valuable course among different texts whose remedy of the category theorem for compact surfaces is both too formalized and intricate for these with out distinctive heritage wisdom, or too casual to come up with the money for scholars a accomplished perception into the topic. Its committed, student-centred procedure info a near-complete facts of this theorem, broadly favorite for its efficacy and formal good looks. The authors current the technical instruments had to installation the tactic successfully in addition to demonstrating their use in a basically dependent, labored instance. learn more... The category Theorem: casual Presentation -- Surfaces -- Simplices, Complexes, and Triangulations -- the basic staff, Orientability -- Homology teams -- The category Theorem for Compact Surfaces. The category Theorem: casual Presentation -- Surfaces -- Simplices, Complexes, and Triangulations -- the basic crew -- Homology teams -- The category Theorem for Compact Surfaces

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153–220 4. 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.

Z0 / D 0. 2. For any two plane closed path, 1 W Œ0; 1 ! A2 and 2 W Œ0; 1 ! A2 , for every homotopy, F W Œ0; 1 Œ0; 1 ! t; u/, for all t; u 2 Œ0; 1, we have n. 1 ; z0 / D n. 2 ; z0 /. Proof. t; u/ 6D z0 for all t; u, with 0 Ä t Ä 1, 0 Ä u Ä 1. ti ; uj /j < for t 2 Œti ; ti C1 and u 2 Œuj ; uj C1 . ti ; uj / z0 . 3 The Fundamental Group of the Punctured Plane Âi0 Âi D ˇi C1 45 ˇi C k 2 ; P P and we see as before that k D 0. Hence, i Âi0 D i Âi , and we conclude that the index does not change as we pass from uj to uj C1 .

Springer, New York, 1987) 9. S. Massey, A Basic Course in Algebraic Topology, GTM No. 127, 1st edn. (Springer, New York, 1991) 10. R. Munkres, Elements of Algebraic Topology, 1st edn. (Addison-Wesley, Redwood City, 1984) 11. R. Munkres, Topology, 2nd edn. 1 Simplices and Complexes As explained in Sect. 2, every surface can be triangulated. This is a key ingredient in the proof of the classification theorem. Informally, a triangulation is a collection of triangles satisfying certain adjacency conditions.