A Guide to the Classification Theorem for Compact SurfacesSpringer Science & Business Media, 5 de febr. 2013 - 178 pàgines 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-centred 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. |
Continguts
1 | |
Chapter
2 Surfaces | 21 |
Chapter
3 Simplices Complexes and Triangulations | 26 |
Chapter
4 The Fundamental Group Orientability | 37 |
Chapter
5 Homology Groups | 52 |
Chapter
6 The Classification Theorem for Compact Surfaces | 79 |
Appendix
A Viewing the Real Projective Plane in R3 The CrossCap and the Steiner Roman Surface | 104 |
Appendix
B Proof of Proposition 51 | 113 |
Appendix
C Topological Preliminaries | 117 |
Appendix
D History of the Classification Theorem | 150 |
Appendix
E Every Surface Can Be Triangulated | 159 |
Appendix
F Notes | 167 |
173 | |
175 | |
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A Guide to the Classification Theorem for Compact Surfaces Jean Gallier,Dianna Xu Previsualització no disponible - 2015 |
A Guide to the Classification Theorem for Compact Surfaces Jean Gallier,Dianna Xu Previsualització no disponible - 2013 |
Frases i termes més freqüents
1st edn accumulation point affine Ahlfors and Sario Algebraic Topology arcwise connected assume bad segment Brahana canonical cell complex cell complex Chap classification theorem closed path closed set Cohn—Vossen compact surfaces complex with boundary connected sum containing convex cross-caps define Definition denoted disjoint disk edges labeled equivalent Euler—Poincare characteristic f is continuous find first free abelian group function fundamental group geometric realization Given a topological gluing graph homeomorphic homology groups homotopy identified immediately verified infinite intersection inverse isomorphic Jordan Klein bottle Lemma Massey metric space Mobius strip Munkres neighborhood nonempty nonorientable normal form normed vector space notion obtained open ball open cover open set open subset oriented edges p-simplex polygons Princeton projective plane prove Seifert and Threlfall sequence simplex simplices single face sphere Springer step surfaces with boundary Texts in Mathematics theorem for compact topological space torus vertices