Question: Given a smooth compact manifold $M$ without boundary, does $M$
admit a Riemannian metric of positive scalar curvature?
We focus on the case of spin manifolds. The spin structure, together with a
chosen Riemannian metric, allows to construct a specific geometric
differential operator, called Dirac operator. If the metric has positive
scalar curvature, then 0 is not in the spectrum of this operator; this in
turn implies that a topological invariant, the index, vanishes.
We use a refined version, acting on sections of a bundle of modules over a
$C^*$-algebra; and then the index takes values in the K-theory of this
algebra. This index is the image under the Baum-Connes assembly map of a
topological object, the K-theoretic fundamental class.
The talk will present results of the following type:
If $M$ has a submanifold $N$ of codimension $k$ whose Dirac operator has
non-trivial index, what conditions imply that $M$ does not admit a metric of
positive scalar curvature? How is this related to the Baum-Connes assembly
We will present previous results of Zeidler ($k=1$), Hanke-Pape-S. ($k=2$),
Engel and new generalizations. Moreover, we will show how these results fit
in the context of the Baum-Connes assembly maps for the manifold and the
- Geometry and Analysis Seminar