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.

## Publications

## 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

**Prof. Dr. Johannes Ebert**

Project leader

WWU Münster

jeber_02(at)uni-muenster.de

** Thorsten Hertl**

Doctoral student

Georg-August-Universität Göttingen

thorsten.hertl(at)stud.uni-goettingen.de

**Prof. Dr. Thomas Schick**

Project leader

Georg-August-Universität Göttingen

thomas.schick(at)math.uni-goettingen.de

**Prof. Dr. Wolfgang Steimle**

Project leader

Universität Augsburg

wolfgang.steimle(at)math.uni-augsburg.de

**Dr. Vito Felice Zenobi**

Researcher

Georg-August Universität Göttingen

vito.zenobi(at)mathematik.uni-goettingen.de