A manifold is a topological space which is locally homeomorphic to R^n. It is made of open subsets of R^n which are glued together by homeomorphisms. A manifold is said to be smooth if these gluing maps are also diffeomorphisms. Because it looks like R^n near each of its points, a smooth manifold possesses no local invariants. However, its local Euclidean structure allows one to do analysis (differentiation) in small neighborhoods. It also allows one to patch together local data to form global objects such as real-valued functions, vector fields and metrics. These extra structures can, in turn, be used to calculate global invariants of the manifold. (A useful analogy is the use of a basis to calculate the dimension of a vector space.)

For much of the course we will follow the discussion contained in the first three chapters of the book Differential Topology by Guillemin and Pollack. A copy of the book will be placed on reserve so you need not buy it. We will skip some sections of the book and will also cover a few topics which are not discussed there. In the first part of the course we will discuss the basic objects and tools of differential topology, including: topological and smooth manifolds, tangent and vector bundles, immersions and submersions, transversality, homotopy and stability, and Sard's theorem. Then we will discuss Intersection Theory which will allow us to prove several results usually discussed in a first course in algebraic topology such as:

• the Jordan curve theorem (every simple closed curve in R^2 divides R^2 into exactly two regions).
• the Brouwer fixed-point theorem (every self-map of the disc has a fixed-point).
• the Hairy ball theorem (a sphere covered with hair has a bald spot).

If time permits we will discuss other topics such as Morse theory.

Course details

Your grade will be based on five homework sets and an in-class presentation of a more advanced topic which complements the core material and hopefully your personal research interests. The homework sets will be designed to supplement what is covered in class and will consist of both examples and general results. Ideally, you will have chosen a topic for your presentation by mid-October. The presentation should be 35-40 minutes long (= half a class).

My office is MAT 4-103 and my office hours for this course will be Wednesdays 9-10:30. Alternatively, you can email me at ely@math.sunysb.edu to make an appointment.

Homework sheets