# Math 215b

This is a graduate level course on differential topology. It assumes that the students have a good understanding of multivariable calculus (inverse and implicit function theorems, uniqueness and existence results for ODE's, integration of multivariable functions), and some exposure to point set and algebraic topology would be helpful. There will be some review of these so it might be possible to fill in some gaps of knowledge along the way.

There will be two textbooks for the course: Lee's Introduction to Smooth Manifolds, and Robbin-Salamon's online notes Introduction to Differential Topology (available at https://people.math.ethz.ch/~salamon/PREPRINTS/difftop.pdf). The first one is a clear book explaining much of the elementary theory in much more detail than we will have time for in class. It is strongly recommended that you read the relevant portions of Lee's book to complement the lectures. For about half of the duration of the course we will discuss a subset of the topics in the Chapters 5-7 of Robbin-Salamon, loosely following their exposition.

We will have weekly homeworks, and a take home exam at the end of the term. Slightly non-standardly, towards the end of the semester, there will be oral exams (individual), where I will be testing you on your solution sets up to that point. This means that I will ask you questions about what you wrote, with the goal of testing whether you understood what you wrote. If you did not turn in a solution for a problem I will not ask you about that. Your performance on the oral exam will affect your total homework grade.

Grading: 50% homework, 50% final exam

Homeworks and the list of topics covered will be put here weekly.

Week 1: Definition of smooth manifolds, the tangent bundle, applications of inverse/implicit function theorems, partitions of unity, existence of Riemannian metrics, Whitney embedding theorem.

Week 2: Vector fields and their flows, tangent vectors as derivations, Lie bracket, commuting vector fields

Week 3: (two classes) Fiber bundles, vector bundles, normal bundle to a submanifold, statement of tubular neighborhood theorem (proof in Euclidean space)

Week 4: Sard's theorem (no proof), transversality and its consequences, parametric transversality theorem, perturbing sections of bundles (in homework now), bonus lecture (intersection form on 4-manifolds, smooth manifolds vs. topological manifolds)

Week 5: Some linear algebra, differential forms on manifolds, integration of differential forms, interlude: manifolds with boundary, exterior differential, Stokes theorem (Robbin and Salamon, Chapter 5)

Week 6: deRham theory (first pass: Poincare lemma, comparison with singular cochains), Cartan formula and its many consequences, volume forms and orientations

Week 7: (two classes) bonus lecture (symplectic geometry basics), Mayer-Vietoris sequence, good covers.

Week 8: Kunneth formula, Poincare duality, Lefschetz fixed point theorem, Cech-deRham complex (in homework)

(last homework)

Week 9: integration along fibres, Thom isomorphism, intersection theory revisited

Week 10: Euler class, Ehressmann connections, connections on vector bundles