## Mathematics 378, Fall 2008

**The topology of knots and three-manifolds**

This course will occupy the last third of the semester, TuTh 11:40am -
12:55pm in Physics 235. It begins **Tuesday November 4** and runs
through **Thursday December 4**.

Over the past decade or two, there has been an explosion of activity in
the topology of three- and four-manifolds. Results such as the
Poincaré conjecture/theorem have greatly expanded our
understanding of classification questions in the subject. From a
topological perspective, much recent excitement has centered on new
tools such as Seiberg-Witten invariants and Heegaard Floer homology.

This minicourse aims to cover some basic topics in low-dimensional
topology, providing background needed to understand recent
developments in the field.
A main objective will be to introduce
knots, Heegaard splittings of three-manifolds, and Dehn surgery on
knots, culminating in a proof of the Lickorish-Wallace theorem that
all compact orientable three-manifolds are obtained from the
three-sphere by Dehn surgery on some link. In this vein, we will also
discuss Kirby calculus, which indicates when two links yield the same
three-manifold by surgery.
Time permitting, there may be some discussion of
Heegaard Floer homology and its relation to low-dimensional topology.

The level of the course is somewhat flexible but I will try to keep
prerequisites to a minimum. Some basic familiarity with algebraic
topology may be helpful, but a willingness to work with a plethora of
three-dimensional diagrams is probably more important.

You can also download the
course advertisement as a PDF.