Characteristic classes of manifold bundles and graph complexes

The importance of characteristic classes in geometry and topology can hardly be overestimated, but characteristic classes for bundles whose fibers are closed manifolds are still not well understood.

We will break new ground in this direction by introducing new rational characteristic classes of manifold bundles, associated to cycles in a certain graph complex in the sense of Kontsevich.The proposed project is at the forefront of research in rational homotopy theory and its applications to differential topology.

It aims to address and resolve the main questions that were left open by previous work of Ib Madsen and myself on the subject.

The new characteristic classes are a vast generalization of the widely studied Miller-Morita-Mumford classes and they yield new tools to study manifold bundles.

They also provide new tools for attacking long-standing open problems about the homology of graph complexes and automorphisms of free groups. Our approach is based on rational homotopy theory and more specifically a new theory of minimal models for fibrations.

A curious feature of this approach is that it leads to a surprising connection to commutative algebra, via the theory of obstructions to multiplicative structures on minimal free resolutions.The research will be carried out by myself and a PhD student during four years. The project will entail theoretical groundwork as well as extensive calculations and case studies.