Cohomology of automorphisms of manifolds and free groups


Project leader


Funding source

Swedish Research Council - Vetenskapsrådet (VR)


Project Details

Start date: 01/01/2016
End date: 31/12/2019
Funding: 2800000 SEK


Description

The goal of the project is to break new ground in the study of the cohomology of automorphisms of high dimensional manifolds and free groups, by exploiting a new surprising connection between the two fields. Roughly 10 years ago, Madsen and Weiss solved the Mumford conjecture, which concerns the cohomology of automorphisms of surfaces. Since then, efforts have been made to extend Madsen and Weiss' methods to high dimensional manifolds. In the last few years, the study of classifying spaces of automorphisms of high dimensional manifolds (or 'moduli spaces of manifolds') has seen substantial developments along these lines in the work of Galatius, Randal-Williams, Madsen, myself and others. The study of automorphisms of finitely generated free groups goes back to Nielsen and others in the 1920s, but remains an active area of research. Key results on the cohomology of automorphisms of free groups have been obtained by Culler-Vogtmann, Galatius, Hatcher, Kontsevich, Morita, and others, but the cohomology is still largely unknown. Currently, this is a very active field; the development is driven by two groups of researchers in particular: Conant-Hatcher-Kassabov-Vogtmann and Morita-Sakasai-Suzuki. The novelty of the project proposed here is that we connect these two lines of research in a new unexpected way. The key is an observation, due to Madsen and myself, which, in short, says that the cohomology in both cases is governed by the same graph complex. Graph complexes were introduced by Kontsevich in the 90s for the purposes of studying the cohomology of automorphisms of free groups and of moduli spaces of curves. Our discovery is that, quite surprisingly, Kontsevich's graph complex for free groups enters in the study of automorpisms of high dimensional manifolds. Our strategy will be to combine our recent findings with methods from rational homotopy theory, surgery theory and algebraic K-theory of spaces to explore the new possibilities that have arisen. We see the prospect of obtaining significant new results in both areas: In geometry, we expect to obtain a better understanding of the relation between the block diffeomorphisms and the diffeomorphisms of highly connected manifolds, shedding new light on the traditional approach to automorphisms of manifolds. In algebra, we envision a new method for detecting unstable classes in the homology of automorphisms of free groups, potentially giving a new approach to a long-standing open problem known as the Morita conjecture. Furthermore, we see ample opportunities for generalizing our previous results to new classes of manifolds, thereby obtaining new methods for calculating the cohomology of their automorphism groups.


Last updated on 2017-28-07 at 07:50