September 20th: Topological manifolds, partitions of unity, manifold construction lemma, complex projective space, manifolds as quotients
September 23rd: Smooth manifolds, smooth maps and diffeomorphisms, maximal atlas, smooth partition of unity, Whitehead approximation theorem for functions.
September 27thth: Tangent space via derivations.
September 29th: Inverse mapping theorem and implicite function theorem (for manifolds). Examples of submanifolds.
October 4th: SL_nR and SO_n as submanifolds. Manifolds with boundary, including implicite function theorem, and Brouwer's fixed point theorem.
October 6th: Definition of fiber bundles and vector bundles. Vectorbundle construction lemma. Proof that the Hopf fibration and the canonical bundle of CP^n are nor trivial.
October 11th: Examples of vector bundles: tangent bundle, Whitney sum, dual bundle, Alt^k-bundle, etc...
October 14th: prove that for every vector bundle E-> M there is F-> M with E+F -> M trivial, (weak) Whitney embedding theorem, general position theorem
October 18th: prove existence of regular neighborhoods and a couple of applications. Vector fields, Picard Lindeloef, creme anglaise theorem.
Octbober 20th: Thom isotopy extension theorem, normal form for non-vanishing vector fields, Lie brackets and some of its properties. Definition of Lie group.
October 25th: Lie groups, Lie algebra, exponential map, one-parameter subgroups groups, rambling about Lie groups
October 27th: Distributions, Foliations and the Frobenius theorem.
November 8th: Fast review of multilinear algebra, differential forms, uniqueness of exterior derivative
November 9th: Existence of exterior derivative, orientation, existence of volume forms, integration
November 15th: integration of forms, Stokes, Beginning of de Rham.
November 17th: cochain morphisms induce maps on cohomology, functoriality of de Rham cohomology, homotopic cochain morphisms induce the same maps, statement of "homotopic is homotopic" theorem (proof next time) and applications: topological invariance of de Rham cohomology, Poincare lemma, invariane of domain
November 22nd: prove that homotopic maps induce identical maps on de Rham cohomology. Proof of the snake lemma and of Mayer-Vietoris, both for usual de Rham and for the compactly supported one. Calculation of the cohomology of the spheres of complex projective spaces.
November 24th: I rambled a lot about things like the Whitehead manifold. Then discussed the existence of good covers and proved that the Betti numbers of manifolds of finite topology are finite.
November 29th: Proved the Euler formula (that is, Euler charateristic = Euler characteristic) and Poincare Duality.
December 1st: Proof of the Kuenneth formula and discussion of transfer.
December 6th: Proof of the Jordan-Alexander theorem, calculation of Lusternik-Schnirelmann category of CP^n and T^n.
December 8th, 13th and 15th: mod-2-degree, Borsuk-Ulam, degree, intersection form, Thom isomorphism and Thom class, Hopf-index theorem, brief sketch of the proof of the Lefschetz trace formula and Lefschetz fixed point theorem.