# MA6J2 Three-Manifolds

**Not Running in 2015/16**

**Lecturer:**

**Term(s):**

**Status for Mathematics students:** List C

**Commitment:** 30 lectures

**Assessment:** 85% by 3-hour examination 15% coursework

**Prerequisites:** MA222 Metric Spaces and MA3F1 Introduction to Topology

**Leads To: **

**Content:**1) Surfaces, handlebodies, I-bundles, polyhedral

2) Hauptvermutung, Heegaard splittings, S

^{3}, T

^{3}, PHS

3) Reducibility, Alexander's Theorem, knot complements, submanifolds of R

^{3}4) Fundamental group, incompressible surfaces, surface bundles

5) Tori and JSJ decomposition, circle bundles

6) Seifert fibered spaces

7) Loop theorem

8) Normal surfaces

9) Sphere theorem

10) Discussion of geometrization conjecture

Other possible topics:

Poincare conjecture, Fox's reimbedding theorem, space forms spherical, euclidean, hyperbolic, eg dodecahedral space, Thurston's eight geometries, Dehn fillings topologically, algebraically, geometrically, eg fillings of the trefoil, figure eight, non-Haken manifolds, three views of PHS (following Gordon).

**Aims: **An introduction to the geometry and topology of three-dimensional manifolds, a natural extension of MA3F1 Introduction to Topology

**Objectives: **By the end of the module the student should be:

Familiar with the basic examples (the three-sphere, the three-torus, knot components...).

Able to compute the fundamental group of a three-manifold M from a selection of presentations of M.

Familiar with the sphere and torus decomposition.

Able to state the loop theorem and use it (for example, to prove that knot components are aspherical).

**Books:**

*Three-dimensional Topology*by Andrew J Casson

*The Theory of Normal Surfaces*by Cameron Gordon

*Notes on Basic 3-Manifold Topology*by Allen Hatcher

*3-Manifolds*by John Hempel

*Classical Tessellations and Three-Manifolds*by José María Montesinos