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OverviewAt 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. Full Product DetailsAuthor: I.M. Singer , J.A. ThorpePublisher: Springer-Verlag New York Inc. Imprint: Springer-Verlag New York Inc. Edition: Softcover reprint of the original 1st ed. 1967 Dimensions: Width: 15.50cm , Height: 1.30cm , Length: 23.50cm Weight: 0.379kg ISBN: 9781461573494ISBN 10: 1461573491 Pages: 232 Publication Date: 11 July 2012 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: Manufactured on demand  We will order this item for you from a manufactured on demand supplier. Table of Contents1 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.ReviewsAuthor InformationTab Content 6Author Website:Countries AvailableAll regions |
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