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BIRS Workshop Lecture Videos

Obstructions to positive scalar curvature and homotopy invariance via submanifolds of small codimension Schick, Thomas


We want to discuss a collection of results around the following 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 map? 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 submanifold. If time permits and we are in the mood for it, we will also or instead discuss to which extend higher signatures of submanifolds are homotopy invariants, in a context similar to the one above.

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