Office: 4307
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Office Hours: Thursdays 4 - 5 pm
Course description In this course we will study the geometry and topology of 3-manifolds, and related properties of their fundamental groups. Topics include: (1) Basic 3-manifolds and examples, (2) Prime decomposition and JSJ decomposition, (3) Haken 3-manifolds, triangulations and normal surface theory, (4) 3-dimensional geometries and Geometrization, (5) Properties of Hyperbolic 3-manifolds, (6) Computational Aspects: Snappy, Snap, Regina, Twister, (7) Properties of 3-manifold groups, (8) RAAGS, cubical complexes and applications to 3-manifold groups, (9) Recent progress about various long standing conjectures.
Reference Books
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3-manifold topology
- An introduction to 3-manifolds by Jennifer Schultens, AMS-GSM series.
- An introduction to 3-manifolds by Stephan Friedl, available online.
- Notes on basic 3-manifold topology by Allen Hatcher, available online.
- Knots and Links (Chapter 9) by Dale Rolfsen, Publish or Perish
- An introduction to Geometric Topology by Bruno Martelli, available online.
- Geometries of 3-manifolds by Peter Scott, Bulletin of LMS, 15 (1983) online.
- Geometric structures on 3-manifolds by Francis Bonahon, Handbook of Geometric Topology, available online.
3-manifold geometry
I am covering topics from a combination of sources including books and notes above. The main two references are [1] and [5] above. The topic-wise references given below is a good approximation of the topics we are covering in class.
| Date | Topic | References/Reading |
|---|---|---|
| Aug 30 | Preliminaries | Triangulations of manifolds
Triangulation
of surfaces Classification of surfaces: Conway's ZIP proof Zeeman's proof |
| Sept 6 | 3-manifolds, Heegaard Spittings | [1]: 6.1, 6.2, [5]: 11.1 Lens Spaces |
| Sept 13 | Dehn surgery, Irreducibile & Prime 3-manifolds | [1]: 7.1 , [5]: 11.3 Alexander Horned Sphere |
| Sept 20 | Alexander's Theorem, Prime decomposition | [1]: 3.2, 5.5, [5]: 9.2 Morse theory and surfaces |
| Sept 27 | Incompressible surfaces, Haken 3-manifolds | [1]: 3.4, 3.6, [5]: 9.3, 9.4 |
| Oct 4 | Seifert Fibered Spaces | [1]: 3.7 , [5]: Chapter 10 Hopf fibration Video |
| Oct 11 | JSJ decomposition | [1]: 3.8 , [5]: 11.5 Satellite knots and JSJ decomposition |
| Oct 18 | Geometrization of 3-manifolds | Geometrization Wiki |
| Oct 25 | Geometry of Seifert Fibered Spaces | Geometries of 3-manifolds by Peter Scott |
| Nov 1 | Hyperbolic 3-manifolds | |
| Nov 8 | Invariants of Hyperbolic 3-manifolds | SnapPy Demo I Demo II Simplest Hyperbolic Knots |
| Nov 15 | Dehn Surgery again ! | Volumes What is a Hyperbolic 3-manifold ? |
| Nov 22 | No class - Thanksgiving break | |
| Nov 30 | Residual Finiteness of 3-manifold groups | |
| Dec 6 | Separability and LERFness |