tailieunhanh - Papers on Topology

Without much exaggeration, it can be said that only one important topological concept came to light before Poincar¶e. This was the Euler characteristic of surfaces, whose name stems from the paper of Euler (1752) on what we now call the Euler polyhedron formula. When writing in English, one usually expresses the formula as | Papers on Topology Analysis Situs and Its Five Supplements Henri Poincaré Translated by John Stillwell July 31 2009 2 Contents Translator s Introduction 1 Topology before Poincaré. 1 Poincaré before topology. 3 The Analysis situs paper . 5 The five supplements. 7 The Poincare conjecture. 9 Comments on terminology and notation. 11 Acknowledgements. 12 Bibliography . 12 On Analysis Situs 15 Analysis Situs 18 Introduction. 18 1. First definition of manifold. 20 2. Homeomorphism. 22 3. Second definition of manifold. 24 4. Oppositely oriented manifolds. 28 5. Homologies . 30 6. Betti numbers . 30 7. The use of integrals . 32 8. Orientable and non-orientable manifolds. 34 9. Intersection of two manifolds . 40 10. Geometric representation. 49 11. Representation by a discontinuous group. 54 12. The fundamental group . 58 13. Fundamental equivalences. 60 14. Conditions for homeomorphism . 65 15. Other modes of generation . 75 16. The theorem of Euler. 85 17. The case where p is odd. 93 18. Second proof . 95