# Geometric Analysis and Topology Seminar

#### Simply Connected Asymmetric Manifolds

Speaker: Matthias Kreck, Heidelberg

Location: Warren Weaver Hall 1013

Date: Friday, October 13, 2006, 1 p.m.

Synopsis:

Let $$(M, g)$$ be a closed Riemannian manifold, then a symmetry is an isometry $$f$$ from $$M$$ to $$M$$. The Riemannian manifold $$M$$ is called asymmetric if the group of ismmetries is trivial. Now, we forget the metric and pass to differential topology. A closed smooth manifold is called asymmetric if the group of isometries is trivial for all Riemannian metrics on $$M$$. In the seventies many examples of asymmetric manifolds where found, but they were all aspherical or close to aspherical, in particular non simply-connected. This lead to the question posed by Raymond and Schultz in 1976, if there are simply connected aspherical manifolds. Volker Puppe showed in 1994 that this is almost true in the sense that he found simply connected 6-manifolds where the group of orientation preserving isometries is trivial for all metrics $$g$$. Recently I showed that all of his manifolds are actually asymmetric. I will give an introduction to the topic and explain the new results.