Description
Springer-Verlag New York Inc. Lecture Notes on Elementary Topology and Geometry by I.M. Singer J.A. Thorpe
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 doesnt know enough. These notes are an attempt to break up this compartmentalization at least in topologygeometry. 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 Rhams theorem the GaussBonnet 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 subjects application to functional analysis.show more