Flat manifold bundles (i.e. manifold bundles with foliations transverse to the fibers) are classified by homotopy classes of maps to the classifying space of diffeomorphisms made discrete. In this talk, I will talk about homological stability of discrete surface diffeomorphisms and discrete symplectic diffeomorphisms which was conjectured by Morita. I will describe an infinite loop space related to the Haefliger space whose homology is the same as group homology of discrete surface diffeomorphisms in the stable range. Finally, I will discuss some interesting applications to the characteristic classes of flat surface bundles and foliated bordism groups of codimension 2 foliations.
It is possible to have a second talk on Thursday at 14:15-15:00 (Salle à déterminer)
Title : Braid groups and diffeomorphisms of the punctured disk
Abstract: Morita proved that for large enough $g$ the mapping class group of a surface of genus $g$, cannot be realized as a subgroup of the discrete surface diffeomorphism group $Diff(\Sigma_g)$, by showing that there is a homology obstruction. Surprisingly, the situation is different for the braid groups. While braid groups cannot be realized by diffeomorphism groups of punctured disks, as N.\,Salter and B.\,Tshishiku recently showed, we prove that the homology groups of the braid group are summands of the homology groups of the discrete diffeomorphisms of a disk with punctures. This situation is similar to the homeomorphism group of a surface of genus $g>5$ where the mapping class group and the homeomorphism group have the same homology but still there is no section from the mapping class group of such a surface to its homeomorphism groups. Using factorization homology, we also show that there is no homological obstruction to realize surface braid groups by diffeomorphism groups of the punctured surface. We discuss the stable homology of discrete diffeomorphisms of the punctured disk.
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