Ma 151 abc - Geometry and Topology

Part a: Homology Theory. CW complexes, homology and calculation of homology groups, exact sequences, cohomology rings, Poincare duality. Part b: Homotopy Theory and K-theory. Fibrations, higher homotopy groups, and exact sequences of fibrations. Fiber bundles, Eilenberg-MacLane spaces, classifying spaces. K-theory, generalized cohomology theory, Bott periodicity. Part c: Basic Riemannian geometry: geometry of Riemannian manifolds, connections, curvature, Bianchi identities, completeness, geodesics, exponential map, Gauss's lemma, Jacobi fields, comparison theorems, relation between curvature and topology.