Lecture Notes on Elementary Topology and Geometry

I.M. Singer, J.A. Thorpe (Autoren)

Buch | Softcover

232 Seiten

2012 | Softcover reprint of the original 1st ed. 1967
Springer-Verlag New York Inc.
978-1-4615-7349-4 (ISBN)

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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.

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.

Reihe/Serie	Undergraduate Texts in Mathematics
Zusatzinfo	VIII, 232 p.
Verlagsort	New York, NY
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	Algebraische Topologie • Differentialgeometrie • Geometry • Topologie • Topology
ISBN-10	1-4615-7349-1 / 1461573491
ISBN-13	978-1-4615-7349-4 / 9781461573494
Zustand	Neuware