Lecture Notes On Elementary Topology And Geometry

At the present time, the average undergraduate mathematics major finds mathematics heavily compartmentalized. After Calculus, students take courses in analysis and algebra, and depending on their interest, they take courses in special topics. If the student is exposed to topology, it is usually straightforward point set topology; if the student is exposed to geometry, it is usually classical differential geometry. 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, topology and group theory. The material studied includes De Rhams's theorem, the Gauss-Bonnet theorem for surfaces, the functional relation of fundamental group to covering space, and surfaces of constant curvature as homogeneous spaces.