Thomas Schick: Operator algebras and topology: script for the 2001 Trieste summer school on high dimensional manifold theory
- Autor: Thomas Schick
- Titel: Operator algebras and topology (dvi) (pdf)
- in "Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001)", 571--660, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2002; MR 1 937 025 Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001), {http://front.math.ucdavis.edu/math.GT/0209164}
The three lectures on operator algebras and topology introduce a new
set of tools to high dimensional manifold theory, namely techniques
coming from the theory of operator algebras. These are extensively
studies in their own right. We will focus on the basic definitions
and properties, and on their relevance to the geometry and topology
of manifolds.
An central pillar of work in the theory of $C^*$-algebras is the
Baum-Connes conjecture. This is an isomorphism conjecture, as
discussed in the talks of L\"uck, but with a certain special
flavor. Nevertheless, it implies the Novikov conjecture. In the
first talk, the Baum-Connes conjecture will be explained and put
into our context.
One application of the Baum-Connes conjecture is to the positive
scalar curvature question. This will be discussed by Stephan
Stolz. It implies the so called ``stable Gromov-Lawson-Rosenberg
conjecture''. The unstable version of this conjecture said that,
given a closed spin manifolds $M$, a
certain obstruction, living in a certain (topological) $K$-theory
group, vanishes if and only $M$ admits a Riemannian metric with
positive scalar curvature. It turns out that this is wrong, and
counterexamples will be presented in the second talk.
The third talk introduces another set of invariants, also using
operator algebra techniques, namely $L^2$-cohomology, $L^2$-Betti
numbers and other $L^2$-invariants. These invariants, their basic
properties, and the central questions about them, are introduced
in the third talk. This subject is more or less independent from the
other talks of the school.
\textbf{Important notice}: This notes are in a preliminary
form. They haven't gone through proofreading as thorough as the
author would like it to be. It is certain that they will contain
many misprints and typos. Moreover, the presentation at many places
certainly can be improved. They are made public at this stage,
nevertheless, since I feel that they might be useful at least for
some participants of the school. To improve the final version, which
is going to appear in the notes of the school, I would appreciate
any notes and comments, either in person at the school, or via email
schick@math.uni-goettingen.de.
Thomas Schick (schick@math.uni-goettingen.de)