Topology and Geometry in Dimensions 1, 2, and 3

(with a Slant towards Geometric Group Theory)

 Instructor: Kai-Uwe Bux Time: MWF 12:20-13:10 Room: 205 MT Office hours: M 13:15-15:15 Office: 583 MT

Homework

Blurb

Low dimensional objects such as graphs, surfaces, and 3-manifolds have the advantage that we can visualise them. We will study these spaces and the groups that relate to them, in particular the fundamental group and their groups of symmetries.

Possible topics include:

• Dynamics of graph and surface automorphisms: Consider a homeomorphism from a surface to itself. What can we say about the dynamics of iterated applications of this map? For instance, the homemorphism takes curves to curves, and with each iteration, they might become more and more complicated. Do we still obtain something reasonable in the limit?

• JSJ-decompositions of 3-manifolds and finitely presented groups: What happens in the fundamental group of a manifold, when you chop this space up into simpler pieces? You would hope that the fundamental group decomposes, in some sense, into simpler pieces too. Conversely, if you can decompose the fundamental group, can you realise this splitting geometrically?

• Algorithmic problems in low dimensional topology: Someone gives you a knot. Can you decide if that gadget is really knotted? Or someone gives you a surface, can you tell whether it is the torus? Which invariants of topological spaces can be computed?

Prerequisites

This class will be accessible to first year graduate students. Some familiarity with topological spaces and groups, as provided by a undergraduate level topology or algebra class, is welcome.