The tangent bundle of a smooth, compact manifold M is a case of special interest.  If M is the boundary of another manifold W, then integration of tangent bundle characteristic classes over M necessarily yields zero ("Stokes's theorem").  The amazing theorems of Thom (1954) show that the converse is also true; Thom even classifies smooth manifolds "up to cobordism"; that is, modulo boundaries.  The ultimate goal of the course is to prove Thom's theorems.

The first two quarters will cover the classic text Characteristic Classes by John Milnor, with some short sidetrips along the way.  (As a bonus, we will then be in a position to read Milnor's landmark 1956 paper constructing exotic differentiable structures on the 7-sphere.)  Milnor's book covers some of Thom's work, but not the proofs that require hard homotopy theory.  The spring quarter will be devoted to introducing the required homotopy theory, and surveying later developments such as Milnor's calculation of the complex cobordism ring.  It turns out that complex cobordism theory has topological applications that go far beyond its origins in the study of smooth manifolds.

Prerequisite: A year of algebraic topology.

Text: For the first two quarters, the book by Milnor cited above, supplemented with some notes and lectures on principal bundles and "fibre bundles with structure group."  For the spring quarter, course notes and various references to be named later.