Witt groups of complex varieties

The thesis Witt Groups of Complex Varieties studies and compares two related cohomology theories that arise in the areas of algebraic geometry and topology: the algebraic theory of Witt groups, and real topological K-theory. Specifically, we introduce comparison maps from the Grothendieck-Witt and Witt groups of a smooth complex variety to the KO-groups of the underlying topological space and analyse their behaviour. We focus on two particularly favourable situations. Firstly, we explicitly compute the Witt groups of smooth complex curves and surfaces. Using the theory of Stiefel-Whitney classes, we obtain a satisfactory description of the comparison maps in these low-dimensional cases. Secondly, we show that the comparison maps are isomorphisms for smooth cellular varieties. This result applies in particular to projective homogeneous spaces. By extending known computations in topology, we obtain an additive description of the Witt groups of all projective homogeneous varieties that fall within the class of hermitian symmetric spaces.This work was supported by the Engineering and Physical Sciences Research Council [grant number LFAG/067] and by the Cambridge Philosophical Society

The plastikstufe - a generalization of the overtwisted disk to higher dimensions

In this article, we give a first prototype-definition of overtwistedness in higher dimensions. According to this definition, a contact manifold is called "overtwisted" if it contains a "plastikstufe", a submanifold foliated by the contact structure in a certain way. In three dimensions the definition of the plastikstufe is identical to the one of the overtwisted disk. The main justification for this definition lies in the fact that the existence of a plastikstufe implies that the contact manifold does not have a (semipositive) symplectic filling.Comment: This is the version published by Algebraic & Geometric Topology on 15 December 200