Universität Augsburg

Bastien Karlhofer
University of Aberdeen
spricht am
Montag, 6. November 2017
um
16:00 Uhr
im
Raum 2004 (L1)
über das Thema:
Abstract: 
In this talk we shall be interested in the Euler class of oriented, real, flat vector bundles over closed oriented manifolds M and in how to derive bounds and particular simplicial representative cocycles for it only depending on the topology of M. For this we call an oriented, real vector bundle flat, if identifications between its fibers only depend on the homotopy type of a connecting path and so it is induced by a homomorphism from the fundamental group of M into GL_n^+(R). Equivalently being defined as vanishing curvature of some linear connection on the vector bundle, flatness of vector bundles and manifolds has been a broad field of study for a long period of time, with characteristic classes and in particular the Euler class carrying far reaching information about the structure of such a bundle. In fact, in 2 dimensions Milnor showed that an oriented, real vector bundle over a closed oriented surface of genus g > 0 is flat, if and only if the absolute value of its Euler class in the second cohomology is less than g, implying that no surface of genus g > 1 is a flat manifold. In higher dimensions there is little hope for a full classification of flatness though. Motivated by this we will outline the Sullivan procedure, that still allows us to find bounds of the absolute value of the Euler class of a flat, oriented, real vector bundle over a closed oriented manifold M depending on the number of simplices of a given triangulation of M and use the simplicial volume in order to extract even better bounds. Finally, we will combine this to extract Sullivan cocycles vanishing over certain subspaces. 
Hierzu ergeht herzliche Einladung. 
Prof. Dr. Bernhard Hanke 
Kaffee, Tee und Gebäck eine halbe Stunde vor Vortragsbeginn im Raum 2006 (L1).