Lecture Notes on Elementary Topology and Geometry

J. A. Thorpe

ISBN 10: 1461573491 / ISBN 13: 9781461573494

Editore: Springer New York, Springer New York, 2012

Riguardo questo articolo

Druck auf Anfrage Neuware - Printed after ordering - At the present time, the average undergraduate mathematics major finds mathematics heavily compartmentalized. After the calculus, he takes a course in analysis and a course in algebra. Depending upon his interests (or those of his department), he takes courses in special topics. Ifhe is exposed to topology, it is usually straightforward point set topology; if he is exposed to geom etry, it is usually classical differential geometry. The exciting revelations that there is some unity in mathematics, that fields overlap, that techniques of one field have applications in another, are denied the undergraduate. He must wait until he is well into graduate work to see interconnections, presumably because earlier he doesn't know enough. These notes are an attempt to break up this compartmentalization, at least in topology-geometry. What the student has learned in algebra and advanced calculus are used to prove some fairly deep results relating geometry, topol ogy, and group theory. (De Rham's theorem, the Gauss-Bonnet theorem for surfaces, the functorial relation of fundamental group to covering space, and surfaces of constant curvature as homogeneous spaces are the most note worthy examples.) In the first two chapters the bare essentials of elementary point set topology are set forth with some hint ofthe subject's application to functional analysis.

Codice articolo 9781461573494

Dati bibliografici

• Titolo: Lecture Notes on Elementary Topology and Geometry
• Editore: Springer New York, Springer New York
• Data di pubblicazione: 2012
• Lingua: Inglese
• ISBN 10: 1461573491
• ISBN 13: 9781461573494
• Rilegatura: Taschenbuch
• Condizione: Neu
• Dimensioni: 235 L x 155 A x 14 P (millimetri) 235 millimetri di larghezza per 155 millimetri di altezza per 14 millimetri di profondità
• Peso: 376 grammi

Descrizione articolo

Riassunto

At the present time, the average undergraduate mathematics major finds mathematics heavily compartmentalized. After the calculus, he takes a course in analysis and a course in algebra. Depending upon his interests (or those of his department), he takes courses in special topics. Ifhe is exposed to topology, it is usually straightforward point set topology; if he is exposed to geom­ etry, it is usually classical differential geometry. The exciting revelations that there is some unity in mathematics, that fields overlap, that techniques of one field have applications in another, are denied the undergraduate. He must wait until he is well into graduate work to see interconnections, presumably because earlier he doesn't know enough. These notes are an attempt to break up this compartmentalization, at least in topology-geometry. What the student has learned in algebra and advanced calculus are used to prove some fairly deep results relating geometry, topol­ ogy, and group theory. (De Rham's theorem, the Gauss-Bonnet theorem for surfaces, the functorial relation of fundamental group to covering space, and surfaces of constant curvature as homogeneous spaces are the most note­ worthy examples.) In the first two chapters the bare essentials of elementary point set topology are set forth with some hint ofthe subject's application to functional analysis.

Contenuti

1 Some point set topology.- 1.1 Naive set theory.- 1.2 Topological spaces.- 1.3 Connected and compact spaces.- 1.4 Continuous functions.- 1.5 Product spaces.- 1.6 The Tychonoff theorem.- 2 More point set topology.- 2.1 Separation axioms.- 2.2 Separation by continuous functions.- 2.3 More separability.- 2.4 Complete metric spaces.- 2.5 Applications.- 3 Fundamental group and covering spaces.- 3.1 Homotopy.- 3.2 Fundamental group.- 3.3 Covering spaces.- 4 Simplicial complexes.- 4.1 Geometry of simplicial complexes.- 4.2 Barycentric subdivisions.- 4.3 Simplicial approximation theorem.- 4.4 Fundamental group of a simplicial complex.- 5 Manifolds.- 5.1 Differentiable manifolds.- 5.2 Differential forms.- 5.3 Miscellaneous facts.- 6 Homology theory and the De Rham theory.- 6.1 Simplicial homology.- 6.2 De Rham’s theorem.- 7 Intrinsic Riemannian geometry of surfaces.- 7.1 Parallel translation and connections.- 7.2 Structural equations and curvature.- 7.3 Interpretation of curvature.- 7.4 Geodesic coordinate systems.- 7.5 Isometries and spaces of constant curvature.- 8 Imbedded manifolds in R3.