Math 215C - Evolving Syllabus

DateMaterial covered
4/4 No lecture.
4/6 Introduction. Naturality properties of cup product and cap product. On a compact, oriented n-manifold, the cup product pairing is non-degenerate (induces an isomorphism from Hk modulo torsion to Hom(Hn-k,Z)).
4/9 Poincare duality for manifolds with boundary and some applications: non-degenerate pairing on the middle homology of an oriented even-dimensional compact manifold without boundary.
4/11 Algebraic structure of the cup product on the middle cohomology of an even dimensional manifold which is the boundary of another compact manifold. Technical results about (topological) manifolds: Existence of collar neighborhoods of the boundary. Any (compact) topological manifold is homeomorphic to a subspace of euclidean space, and is the retract of a neighborhood there.
4/16 Smooth manifolds. Motivation and basic definitions: Atlas, smooth atlas, smooth structure, smooth manifold. Smooth functions between such. Submanifolds and the implicit function theorem. (Also mentioned the Brieskorn spheres.)
4/18 Smooth vector bundles on a smooth manifold. The tangent bundle of a smooth manifold. Tangent vectors as differential operators on germs of functions. Vector fields as differential operators on functions. Transversality (in general for two smooth functions with the same target; as a special case, a map can be transverse to a submanifold; two submanifolds can be transverse). If f0 and f1 are transverse, then the pullback is a smooth submanifold of M1 x M2.
4/23 Whitney embedding theorem. Class taught by Ralph Cohen.
4/25 Sard's theorem. Whitney embedding theorem: bound on dimensions, and non-compact manifolds.
4/30 Thom's transversality theorem.
5/2 More on Thom's transversality theorem.