These are a slightly revised version of the course notes that were distributed
during the course on Geometry of 3-Manifolds at the Tur´an Workshop on Low
Dimensional Topology in Budapest, August 1998.
The lectures and tutorials did not discuss everything in these notes. The notes
were intended to provide also a quick summary of background material as well as
additional material for “bedtime reading.” There are “exercises” scattered through
the text, which are of very mixed difficulty. Some are questions that can be quickly
answered. Some will need more thought and/or computation to complete. Paul
Norbury also created problems for the tutorials, which are given in the appendices.
There are thus many more problems than could be addressed during the course,
and the expectation was that students would use them for self study and could ask
about them also after the course was over.
For simplicity in this course we will only consider orientable 3-manifolds. This
is not a serious restriction since any non-orientable manifold can be double covered
by an orientable one.
In Chapter 1 we attempt to give a quick overview of many of the main concepts
and ideas in the study of geometric structures on manifolds and orbifolds in dimension 2 and 3. We shall fill in some “classical background” in Chapter 2. In Chapter
3 we then concentrate on hyperbolic manifolds, particularly arithmetic aspects.
Lecture Plan:
1. Geometric Structures.
2. Proof of JSJ decomposition.
3. Commensurability and Scissors congruence.
4. Arithmetic invariants of hyperbolic 3-manifolds.
5. Scissors congruence revisited: the Bloch group