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MA4J2 Three-Manifolds

Not Running in 2019/20



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:

1) Surfaces, handlebodies, I-bundles, polyhedral
2) Hauptvermutung, Heegaard splittings, S3, T3, PHS
3) Reducibility, Alexander's Theorem, knot complements, submanifolds of R3
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).

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


Additional Resources