Oberwolfach Seminar: The Novikov Conjecture: Geometry and Algebra
- January 25th - 31st, 2004
- Deadline for applications:
- December 1, 2003
- Matthias Kreck, Heidelberg
- Wolfgang Lück, Münster
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.
(Co)Homology, homology, homotopy groups, manifolds and
vector bundles, characterstic classes, bordism.
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).
Mathematisches Forschungsinstitut Oberwolfach
updated: August 15th, 2003