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MA4J2 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, 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 (S3, ∏3, knot components...)
Able to compute ∏, (M3) from a variety of presentations of M.
Familiar with the sphere and torus decomposition.
Able to state the loop theorem and use it (e.g. 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

 

Additional Resources

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Year 1 regs and modules
G100 G103 GL11 G1NC

yr2.jpg
Year 2 regs and modules
G100 G103 GL11 G1NC

yr3.jpg
Year 3 regs and modules
G100 G103

yr4.jpg
Year 4 regs and modules
G103

Archived Material
Past Exams
Core module averages