Differential topology and de Rham cohomology
I will only assume a decent base of calculus in several variables (being scared of the implicit function theorem is ok, being so scared that one runs away or it or stops listening/reading/thinking is not), a decent basis of linear algebra (mostly multilinear algebra), and some contact with pointwise topology (what is a topology, what means Hausdorff, compact, connected and such). That being said, I will go relatively fast. Besides introducing the main players (manifolds and bundles) I will spend some time talking about things like the Frobenius theorem or about Lie groups (proving the bijection between Lie subgroups of a Lie group and Lie subalgebras of its Lie algebra, and hopefully also proving Cartan's closed subgroup theorem). That is basically the first part of the course.
The second part is about de Rham cohomology. Although there are surely people who have seen it before, I will prove Stokes' theorem. No extra prerequisites are needed for this, meaning that I will discuss all the needed algebra. The goal is to do things like the topological invariance of de Rham cohomology, invariance of domain, Poincare Duality, the Kuenneth formula, Jordan-Alexander duality, the Thom isomorphism and the Lefschetz fixed point theorem (maybe the Lefschetz trace formula). Intermixed with this, I will discuss a minimum of mod2 and integral intersection theory, that is things like the relation between the intersection of submanifolds of complementary dimension and of their Poincare duals. (When I type this here, it seems like a lot, but we will see…)
There are plenty of good books, and I will mention them in class, but I will provide note and at least for part of the class, Lee's book Introduction to Smooth manifolds is very good. A jewel that everybody should read is Milnor's book Topology from the Differentiable Point of View.- Week 1: Topological manifolds, C^0-partitions of unity, submanifolds, manifold construction lemma, projective space, manifolds as quotients, smooth manifolds and smooth maps, composition of smooth maps is smooth
- Week 2: smooth partitions of unity, smooth manifold construction lemma, smooth manifolds as quotients, smooths manifolds as submanifolds, tangent space and differential, inverse mapping theorem, implicit function theorem
- Week 3: SL_nR as a submanifold, submersions, immersions, embeddings, every manifold arises as a submanifold of some euclidean space, Brouwer fixed point theorem, fiber bundles, Hopf fibration, Hopf fibration is non-trivial.
- Week 4: Canonical bundle over CP^n, tangent bundle, Whitney sum, dual bundle, pull-back, normal bundle... for every E->M there is F->M with E+F trivial. Extension of sections.
- Week 5: Regular neighborhoods, Whitney approximation, nearby maps are homotopic, weak Whitney embedding theorem, general position, closed co-dimension one submanifold of R^n separate. Picard-Lindeloef for open sets in R^n.
- Week 6: Picard-Lindeloef and flows for manifolds. Isotopy extension theorem. Lie bracket. Flows commute if and only if Lie bracket vanishes. Vector fields can be made coordinate vector fields if and only if Lie bracket vanishes. Distributions of k-planes, and distribution associated to a foliation.
- Week 7: The Frobenius theorem, Lie groups (Lie algebra, exponential, Lie group homomorphisms, bijection between Lie subgroups and Lie subalgebras, consequences of Cartan's closed subgroup theorem)
- Week 8: Differential forms, exterior differential, orientability, integration of top-degree forms, Stokes' theorem, another proof of the fixed point theorem
- Week 9: cochain complexes, cohomology, de Rham and compactly supported, calculation of H^*(R^1) and H^*1(S^1), cochain homotopies, homotopy invariance of de Rham cohomology, topological invariance, Poincare lemma, dimension invariance, snake lemma, statement of Mayer-Vietoris
- Week 10: Proof of Mayer-Vietoris, calculation of cohomology of sphere and CP^n, Jordan-Alexander duality, finite dimensionality, beginning of proof of Euler-formula (discussion of what is a triangulation)
- Week 11: Proof of Euler-formula (that was a flop), Poincare duality, corollaries of Poincare duality, proof of Poincare lemma for compactly supported cohomology, the Kuenneth formula, Poincare polynomials, mod(2)-degree is well-defined.