09

Diffeomorphisms and the topology of positive scalar curvature

One of the fundamental goals of geometric topology is to understand the interplay between an underlying topology and the types of geometry supported by this topology.

For this project, the "underlying topology" is given by a smooth manifold $$M$$, and also the diffeomorphism group of such a manifold is part of this "underlying topology". A lot of our knowledge about the diffeomorphism group uses homotopy theory. One part of our project will be concerned with explicit geometric constructions of this homotopy theoretically given information. We then aim at exploiting this for new (secondary) constructions and applications.

The type of applications we plan to concentrate on is the study of the action of the diffeomorphism group on the space of metrics of positive scalar curvature $${\mathcal R}^+(M)$$ on $$M$$. Such applications raise the following research challenges and questions:

• get appropriate information about $$Diff(M)$$. Classically, this involves unstable homotopy theory, smoothing theory and other tools from algebraic and geometric topology.
• find tools to understand the effect of the action on $${\mathcal R}^+(M)$$, and apply them to the situation at hand. This will involve index theory of Dirac operators and the Gromov-Lawson surgery technique for metrics of positive scalar curvature.
• develop concordance space variants for $${\mathcal R}^+(M)$$ which allow for more flexible constructions and are more appropriate for calculations, and compare them to the true $${\mathcal R}^+(M)$$.
• find explicit geometric constructions. We do this to get a better theoretic understanding of the geometric content of the abstract homotopy theoretic information about $${\mathcal R}^+(M)$$, and also with the hope to arrive at new and additional tools.
• develop rigidity results on the action of $${\mathcal R}^+(M)$$. It is known that in many cases this action factors through a cobordism category. Computation in the kernel of the passage from $$Diff(M)$$ to this cobordism category should show that large subgroups of the homotopy groups of $$Diff(M)$$ act trivially on the homotopy groups of $${\mathcal R}^+(M)$$.
• Finally, we will adress the question to which extent the action of (compactly supported) diffeomorphisms on a non-compact manifold $$M$$ gives rise to non-trivial classes in $${\mathcal R}^+(M)$$, where $${\mathcal R}^+(M)$$ now denotes the space of complete uniformly positively scalar curvature metrics. This will involve an interplay between the flexibility given by "pushing off to infinity" and new tools from coarse index theory.

## Team Members

Dr. Simone Cecchini
Researcher
Georg-August Universität Göttingen
simone.cecchini(at)uni-goettingen.de

Prof. Dr. Diarmuid Crowley
Principal investigator
University of Melbourne
dcrowley(at)unimelb.edu.au

Thorsten Hertl
Doctoral student
Georg-August-Universität Göttingen/Universität Augsburg
thorsten.hertl(at)stud.uni-goettingen.de

Prof. Dr. Thomas Schick