Introductory description

Topology is the study of what remains of an object X once we forget distances and angles, but remember "closeness". Amazingly we sometimes can, from the topology of X alone, recover the collection of best geometries on X. This gives rise to the "moduli space of geometries on X". The study of such moduli spaces lies at the heart of geometric topology. There are two outstanding examples:
(a) the moduli space of hyperbolic geometries on a surface is a variety of complex dimension 3g - 3 (where g is the genus) first studied by Riemann and Klein;
(b) the moduli space of hyperbolic geometries on a three-manifold is (at most) a point, as proven by Mostow. These two behaviours - flexibility in dimension two and rigidity in dimension three - give geometric topology its unparalleled richness.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Again this will vary depending on the overarching theme chosen. Here
are syllabi for the three representative themes.

A. Knot theory: Basic examples, knot diagrams. Reidemeister's theorem. Knot polynomials and the skein relation. Seifert genus and Seifert's algorithm. Sutured manifolds following Scharlemann. Proof of the hierarchy theorem. Disk and sphere theorem for Haken manifolds, knot complements are aspherical. Hyperbolic geometry, Thurston's
classification of knots, volumes of knots.

B. Mapping class groups: Examples and constructions of surfaces, orbifolds, and homeomorphisms. Essential curves and geometric intersection number. The mapping class group. Periodic classes and Nielsen realisation. Dehn twists. The curve complex is connected.
Dehn twists (finitely) generate the mapping class group. Train tracks and measured laminations. Pseudo-Anosov homeomorphisms. The Nielsen--Thurston classification. PML is a sphere. Teichmuller space. The Thurston compactification. Proof of the NT classification. Subsurface projections. Hierarchies and the Masur--Minsky distance estimate.

C. Three-manifolds: Examples and constructions. The Hauptvermutung: every three-manifold admits a triangulation. The prime decomposition theorem. The sphere, disk, torus, and annulus theorems:
three-manifolds are (essentially) determined by their fundamental groups. JSJ theory: splittings of three-manifold fundamental groups along abelian subgroups are canonical. Seifert fibered spaces. Mapping class groups of surfaces and surface bundles. The eight Thurston geometries. Mostow rigidity for finite volume hyperbolic three-manifolds.

Learning outcomes

By the end of the module, students should be able to:

• A mastery of the basic examples, an ability to compute the basic invariants, and an appreciation of the outstanding structural theorems in the field.

Indicative reading list

A. Knot theory: Purcell - Hyperbolic Knot Theory; Scharlemann -
Lectures on the theory of sutured manifolds; Thurston -
Three-dimensional Geometry and Topology;

B. Mapping class groups: Casson and Bleiler - Automorphisms of surfaces
after Nielsen and Thurston; Farb and Margalit - The primer on mapping
class groups.

C. Three-manifolds: Hempel - 3-manifolds; Hatcher - Notes on Basic
3-Manifold Topology; Casson - Three-dimensional topology; Thurston -
Three-dimensional Geometry and Topology.

Subject specific skills

A. Knot theory: Given a knot diagram of a knot K find (a) a
presentation of the fundamental group, (b) a Seifert surface, (c) a
sutured manifold hierarchy for S^3 - K, (d) the Seifert genus of K,
(e) the Thurston type of K, and (f) an estimate for the hyperbolic
volume of S^3 - K (in the hyperbolic case).

B. Mapping class groups: Given a train-track on a surface determine
its (a) combinatorial invariants (orientability, numerical stratum,
spin) and (b) its polytope of carried curves. Given a homeomorphism
(as a product of twists) find its invariant train track (if any) and
determine its Nielsen--Thurston type.

C. Three-manifolds: Given a triangulation of a three-manifold M find
(a) a presentation of the fundamental group, (b) the homology groups,
(c) the prime decomposition, (d) the JSJ decomposition, (e) the
Seifert invariants (if M is Seifert fibered), and (f) the Thurston
geometry of M (if it exists). Similarly, given a knot diagram D for a
knot K, find the above for the manifold M_K = S^3 - K.

Transferable skills

• sourcing research material
• prioritising and summarising relevant information
• absorbing and organizing information
• presentation skills (both oral and written)