The mapping class group
and the curve complex, Term I, 2008
This course will be an elementary introduction to curves on
surfaces. Following the work of Masur and Minsky we will investigate
the complex of curves. The structure of the complex of curves
is hierarchical and tightly interlocked with the structure of the
mapping class group, the Teichmuller space and Kleinian groups.
We will start with the very basic notions of surface topology and
develop the necessary tools in geometry and algebra as the term
Topics may include: curves on surfaces, isotopy classes, the bigon
criterion, the complex of curves C(S), curve surgery, connectedness of
C(S), distance in C(S), projectively measured laminations, Dehn
twists, mapping classes (Nielsen-Thurston classification), generation
of the mapping class group by twists (Lickorish's theorem),
combinatorial rigidity of the curve complex (Ivanov's theorem), coarse
geometry of the curve complex (Masur-Minsky distance estimate).
The schedule has a list of topics,
organized by week. These are subject to change, depending on the
whims of the instructor and of the class.
||s.schleimer at warwick dot ac dot uk
||024 7652 3560
||WThF 10-11am and by appointment.
Please do ask questions in class. I am also more than happy to
answer questions via email or in office hours.
As the course progresses I hope to revise my "Notes on the complex
of curves", available on my research page. However, students are advised to
read some of the primary material:
- Curves on 2-manifolds and isotopies, by Epstein.
- A representation of orientable combinatorial 3-manifolds, by Lickorish.
- A finite set of generators for the homeotopy group of a 2-manifold, by Lickorish.
- On the geometry and dynamics of diffeomorphisms of surfaces, by Thurston.
- Boundary structure of the modular group, by Harvey.
- Automorphism of complexes of curves and of Teichmuller spaces, by Ivanov.
- Automorphisms of complexes of curves on punctured spheres and on punctured tori, by Korkmaz.
- Automorphisms of the complex of curves, by Luo.
- Geometry of the complex of curves I and II, by Masur and Minsky.
Other works of interest include:
ZIP proof, by Francis and Weeks
- Braids, links, and mapping class groups, by Birman.
- The geometries of 3-manifolds, by Scott.
- Automorphisms of Surfaces After Nielsen and Thurston, by Casson and Bleiler.
- Knots, links, braids and 3-manifolds, by Prasolov and Sossinsky.
- Three-dimensional Geometry and Topology, by Thurston and edited by Levy.
One homework exercise per 1.5 weeks. (Exercises can be found in the
notes and will also be stated in lecture.) Email solutions to the
Please tell me in person, or via email, about any errors on this
website or made in class. I am especially keen to hear about
mathematical errors, gaffes, or typos appearing in my notes on the