Vendredi 28 avril 2017

Le neuvième Karlsruhe-Heidelberg-Strasbourg "Geometry Day" aura lieu à Strasbourg le 28 avril 2017.

  • Vendredi 28 avril 2017

  • 12:00

    Cubical geometry in the polygonalisation complex

    — Valentina Disarlo (Universität Heidelberg)

    We introduce the polygonalisation complex of a surface, a cube complex whose vertices correspond to polygonalisations. This is a geometric model for the mapping class group and it is motivated by works of Harer, Mosher and Penner. Using properties of the flip graph, we show that the midcubes in the polygonalisation complex can be extended to a family of embedded and separating hyperplanes, parametrised by the arcs in the surface. We study the crossing graph of these hyperplanes and prove that it is quasi-isometric to the arc complex. We use the crossing graph to prove that, generically, different surfaces have different polygonalisation complexes. The polygonalisation complex is not CAT(0), but we can characterise the vertices where Gromov’s link condition fails. This gives a tool for proving that, generically, the automorphism group of the polygonalisation complex is the (extended) mapping class group of the surface.
  • 14:30

    The signature modulo 8 of surface bundles over surfaces

    — Caterina Campagnolo (Karlsruhe Institute of Technology)

    Joint work with Dave Benson, Andrew Ranicki and Carmen Rovi The signature is an important invariant of manifolds. It has being widely studied for bundles, where a fundamental result is Chern, Hirzebruch and Serre’s sufficient condition for its vanishing. Meyer showed in 1973 that the signature of surface bundles over surfaces is a multiple of 4 and that it can be computed using a degree two cohomology class of the symplectic group with integral coefficients. Turaev later found an explicit cocycle for the generator of this cohomology group. This being understood, a natural task is to study signature modulo 8. Several questions arise : is there a Chern-Hirzebruch-Serre result in the reduced setting? Does there exist a Meyer class computing the signature modulo 8? Which is the smallest group detecting such an invariant ? I will present the context and our joint work around this subject.
  • 16:00

    The IHP Collection (colloquium)

    — François Apéry Oliver Labs (UHA/MO-Labs)