Summer School

Low Dimensional Topology

Budapest, Hungary, August 3 - 14, 1998

A satellite summer school of the

International Congress of Mathematicians

SUMMARY of the topics

In the last years, the study of low--dimensional spaces became one of the most important subjects of the mathematics. In this Summer School, we present some important aspects of the geometry and topology of low--dimensional manifolds. Moreover, we emphasize several connections with other field of mathematics. The first connection deals with links of algebraic germs of singular surfaces, the second with the geometric group theory, i.e. the study of those groups which occur as fundamental groups of compact nonpositively curved spaces.

The four series of lectures will cover the following topics:

Prof. John W. Morgan (``Differential topology of 4--dimensional manifolds'') will explain the new, famous invariants of differentiable 4--manifolds (namely, the invariants provided by Donaldson's and Witten's theory), and will present some classification theorems in differential topology.

The series of lectures ``Geometry of 3--manifolds'' of Prof. W. D. Neumann will start with a new short proof of the toral decompositon theorem, then survey the current state of Thurston's geometrization conjecture, which says that the components of the toral decompositon of a three-manifold should be geometric. The bulk of the lecture will then deal with the most interesting case of geometric 3-manifolds -- the hyperbolic ones, describing some old and some new results.

The lectures of Prof. A. Némethi (``The link of surface singularities'') will focus on those three--manifolds which are links of germs of algebraic surface singularities. They decompose in Seifert manifolds, and can be represented by plumbing diagrams. The goal of the lecture is a hunting of connections between the link--invariants and the smoothing invariants (or even more sophisticated analytic invariants) of the singular germs of spaces.

The talks of Prof. M. Davis (``Nonpositively curved spaces'') present the increasingly important area of geometric group theory, following the work of Gromov. As Aleksandrov showed, the notion of nonpositive curvature can be defined for a more general class of metric spaces than Riemannian manifolds. One of the basic results in this area is a generalization of the Cartan--Hadamard Theorem: if a nonpositively curved space is complete and simply connected then it is contractible. It follows that if a nonpositively curved space is compact, then it is an Eilenberg--MacLane space of type K(G,1) where G denotes its fundamental group. Nonpositively curved polyhedra occur naturally in mathematics as spaces asssociated to reflection groups and to Tits buildings, as convex hypersurfaces in Minkowski space, as blow-ups of real projective space along certain subpaces, and a variety of other contexts. The connection with low--dimensional topology and hyperbolic geometry will be emphasized.