Thomas Schick: Index theory and the Baum-Connes conjecture
These notes are based on lectures on index theory, topology, and operator
algebras at the ``School on High Dimensional Manifold Theory'' at
the
ICTP in Trieste, and at the Seminari di Geometria 2002 in
Bologna. We describe how techniques
coming from the theory of operator algebras, in particular
$C^*$-algebras, can be used to study manifolds. Operator algebras are extensively
studied in their own right. We will focus on the basic definitions
and properties, and on their relevance to the geometry and topology
of manifolds. The link between topology and analysis is provided by
index theorems. Starting with the classical Atiyah-Singer index
theorem, we will explain several index theorems in detail.
Our point of view will be in particular, that an index lives in a
canonical way in the K-theory of a certain $C^*$-algebra. The
geometrical context will determine, which $C^*$-algebra to use.
A central pillar of work in the theory of $C^*$-algebras is the
Baum-Connes conjecture. Nevertheless, it has important direct applications to the
topology of manifolds, it implies e.g.~the Novikov conjecture. We
will explain the Baum-Connes conjecture and put it
into our context.
Several people contributed to these notes by reading preliminary
parts and suggesting improvements, in particular Marc Johnson, Roman
Sauer, Marco Varisco und Guido Mislin. I am very indebted to all of
them. This is an elaboration of the first chapter of the author's
contribution to the proceedings of the above mentioned ``School on High
Dimensional Manifold Theory'' 2001 at the ICTP in Trieste.
Thomas Schick (schick@math.uni-goettingen.de)