Fall 2002 : MAT 566 Differential Topology

Tuesday and Thursday 9:50-11:10 PHY P123

Introduction to differential topology

Differential topology is a subject in which geometry and analysis is used to obtain topological invariants of spaces, often numerical. Some examples are the degree of a map, the Euler number of a vector bundle, the genus of a surface, the cobordism class of a manifold (the last example is not numerical).

A manifold is a topological space which is locally homeomorphic to R^n. It is made of open subsets of R^n glued together by homeomorphisms. If these gluing maps are diffeomorphisms, then we obtain a smooth (or differentiable) manifold. These are the basic objects of study of differential topology. They are also the basic objects of study of differential geometry, but whereas differential geometry is concerned mainly with local invariants (for example, curvature), differential topology is concerned with global issues.

Manifold occur in many branches of mathematics, for example as Lie groups in algebra, as space-time in relativity, as phase-space in mechanics, as state-space in dynamics and differential equations. An important idea in differential topology is the passage from local to global information. As an illustration of the distinction consider differential equations : two solutions which may look similar locally can turn out to have very different global properties (one may be periodic, the other convergent).

In the above examples there is typically additional structure, such as a group structure (multiplication and inverses), a metric, or a symplectic structure. In differential topology we are interested in the manifold itself and the additional structure is just a tool. This is similar to algebraic topology, which employs triangulations and uses combinatorial methods to obtain topological information.

As an example, let's look at the Euler characteristic of the sphere S^2. In algebraic topology we have a combinatorial definition. We take a triangulation, and then count the number of vertices, minus the number of edges, plus the number of faces, to get two. In the case of a Platonic solid this is just Euler's theorem. In differential topology we count the number of zeros of a smooth vector field, weighted by their indices, and once again get two. There can be no non-vanishing smooth vector field, a fact known as the hairy ball theorem. Finally, a more geometric definition requires us to choose a metric on the sphere then take the Gaussian curvature K of the corresponding connection. Integrating over the sphere we one again get two (up to a factor of 2Pi). This is the Gauss-Bonnet formula. In all of these definitions we employ some additional structure (a triangulation, a vector field, a metric) but ultimately define a topological invariant which is independent of the choices made.

From a topological point of view, manifolds have no local invariants. They look the same at every point, and there are symmetries taking a given point to any other in the same connected component. The kinds of questions that one asks in differential topology are therefore global. For example, can we embed one manifold M in another N? If M is homeomorphic to N, is it diffeomorphic to N? Given M, does there exist N such that M is the boundary of N?

About the course

In this course I hope to cover the following material : topological and smooth manifolds, submanifolds and embeddings, tangent and vector bundles, regular and critical values of maps, Sard's theorem and Brown's theorem, degrees of maps, vector fields and dynamical systems, homotopy and isotopy, connect sum of manifolds, tubular neighbourhoods and cobordisms, manifolds with boundary, tranversality, Morse theory. Depending on time, some additional topics may be covered or some topics may be left out. I'm happy to consult with the class to find out your preferences.

Prerequisites are point-set topology and real analysis. Some examples may occasionally use complex analysis.

The following reference books have been placed on two-hour loan in the Math/Physics/Astronomy Library :

John W. Milnor "Topology from the differential viewpoint" PUP 1965.

Theodor Broecker and Klaus Janich "Introduction to differential topology" CUP 1982.

Morris Hirsch "Differential topology" Springer.

There won't be an examination for this course, but instead I will set about five problem sheets. These will be designed to supplement what is covered in class and will consist of both examples and general results. I strongly urge you to have a look at all of the problems : some of them may contain results which will be important in later lectures, and of course a wide collection of examples is essential to a good understand of the subject.

My office is MAT 4-104 and you can email me on sawon@math.sunysb.edu to make an appointment.

Note: If you have a physical, psychological, medical or learning disability that may impact on your ability to carry out assigned course work, I would urge that you contact the staff in the Disabled Student Services office (DSS), Room 133 Humanities, 632-6748/TDD. DSS will review your concerns and determine, with you, what accommodations are necessary and appropriate. All information and documentation of disability is confidential.

Exercise sheet one: postscript, pdf.

Please attempt four or five problems and submit your work to me at the class on Thursday 3rd October.

Exercise sheet two: postscript, pdf.

Please attempt four or five problems and submit your work to me at the class on Thursday 17th October.

Exercise sheet three: postscript, pdf.

Please attempt three or four problems and submit your work to me at the class on Thursday 31st October.

Exercise sheet four: postscript, pdf.

Please attempt three or four problems and submit your work to me at the class on Thursday 14th November.

Exercise sheet five: postscript, pdf.

Please attempt three or four problems and submit your work to me at the class on Thursday 5th December.

Back to the main page.

This page last modified by Justin Sawon
Thursday, 14-Nov-2002 13:12:53 EST
Email corrections and comments to sawon@math.sunysb.edu