Oberwolfach Seminar: The Novikov Conjecture: Geometry and Algebra

Date:

January 25th - 31st, 2004

Deadline for applications:

December 1, 2003

Organizers:

Matthias Kreck, Heidelberg

Wolfgang Lück, Münster

Subjects:

The original Novikov conjecture says that the higher signatures, which are certain numerical invariants of smooth manifolds, are homotopy invariants. It was formulated by Novikov in the sixties and remains one of the challenges in higher-dimensional differential topology. If the universal covering is contractable (a K(pi,1)-manifold), the Novikov conjecture is a consequence of the Borel conjecture saying that such manifolds are homeomorphic if they are homotopy equivalent. The natural approach to the Novikov conjecture uses surgery, a method which studies the difference between homotopy equivalences and s-cobordisms which by Smale's famous theorem are closely related to homeomorphisms or diffeomorphisms. After a definition of the basic invariants, Smale's theorem and surgery, we will define the so-called assembly map, whose injectivity implies the Novikov conjecture. As an illustration of the methods of proof we will go through the case of abelian fundamental groups in detail. Finally we will give a survey about the most recent developments concerning other assembly maps, the conjectures of Baum-Connes and Farrell-Jones and their geometric implications.

Prerequisites:

(Co)Homology, homology, homotopy groups, manifolds and vector bundles, characterstic classes, bordism.

Literature:

The most difficult input is surgery theory. As a start, we suggest to look at the wonderful paper of Kervaine and Milnor: Groups of homotopy spheres, Annals of Math., 77, 504 - 537 (1963). We also recommend W. Lück, A basic introduction to surgery theory, available at the homepage of W. Lück in Münster. A good survey for the Novikov conjecture are the two volumes: Novikov Conjectures, Index Theorems and Rigidity I, II, edited by Steven Ferry, Andrew Ranicki and Jonathan Rosenberg, London Mathematical Society Lecture Notes Series 227, (1993).