Basic course information
Time: MWF 1:00 p.m.
Place: Gerlinger 242
Office hours: by appointment.
Final exam: none.
Prerequisites
Some basic algebraic topology (homology, fundamental group, covering spaces) -- Math 635 suffices -- and some basic knowledge of smooth manifolds. Students outside the mathematics Ph.D. program must obtain the instructor's permission to enroll in this course.
Description and goals
The goal is to introduce some basic tools for studying manifolds in dimensions 2, 3, and 4, including the mapping class groups of surfaces, the moduli space of hyperbolic metrics on a surface, Heegaard splittings and surgery presentations for 3-manifolds, handle decompositions for 4-manifolds, and geometric structures on 3-manifolds. The course is based on lecture notes of Steve Kerckhoff, from courses in 2004, 2015, and 2021.
Course requirements and policies
The course has two tracks, with different requirements. Students must choose a track in Week 1.
Homework track
The course will have three problem sets of at least ten problems each. On the homework track, you must turn in solutions to five problems from each problem set, by the end of weeks 4, 7, and 10.
Mini-paper track
On the mini-paper track, you will write two short papers giving detailed proofs of results where I sketch the proof in class. I will give a list of results you can choose from, updated periodically during the quarter. Different students will write up different results. The goal is to eventually have a library of handouts that students can consult for details on some of the fiddlier points; so, your papers will eventually be public, though you can choose not to have your name attached to them if you prefer. Writing the paper involves understanding the argument from class, giving a little background, and synthesizing material from some books or papers to fill in the details.
On this track, in addition to writing the two papers, you will also give feedback and suggestions on three papers other students write, and will revise your papers to take into account feedback from other students and me.
All the papers will be managed in a shared git repository, so you will also learn how to use git for collaboration on papers. (I will provide the repository, templates for papers, and instructions.)
List of possible topics (will expand over time)
Academic honesty
In both the homework and mini-paper track, you are welcome to get help from me, printed references like books and papers, electronic resources like lecture notes posted to the web or comments on MathOverflow, and other students. However, all sources from any of these forms must be cited fully. You must write up your solutions or paper on your own. If you have questions about how or when to cite a source, ask me. Failure to cite sources is academic dishonesty and will be reported.
Students with disabilities
I, and the University of Oregon in general, are committed to an inclusive learning environment. If you have a disability which may impact your performance on exams, please contact the Accessible Education Center to discuss appropriate accommodations. If there are other disability-related barriers to your participation in the course, please either discuss them with me directly or consult with the Accessible Education Center.
Homework
- Homework 1. Due April 27.
- Homework 2. Due May 18.
- Homework 3. Due June 7.
Schedule
| Week | Date | Topic |
| 1 | 4/1 | Course overview. Definition of the mapping class group. Its action on the space of simple closed curves. |
| 4/3 | The mapping class group of the torus. Smale's theorem on the diffeomorphisms of the disk rel boundary. |
|
| 4/5 | Presentation of PSL(2,Z). |
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| 2 | 4/8 | Dehn twists, sketch of Lickorish's theorem. |
| 4/10 | Finish sketch of Lickorish's theorem. |
|
| 4/12 | Proof of Smale's theorem. The diffeomorphism group of the sphere. |
|
| 3 | 4/15 | Heegaard splittings |
| 4/17 | Dehn surgery, Lickorish-Wallace theorem |
|
| 4/19 | Handles, cobordisms, and the relation to surgery |
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| 4 | 4/22 | More on framings, Kirby calculus |
| 4/24 | More Kirby calculus |
|
| 4/26 | More Kirby calculus |
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| 5 | 4/29 | The hyperbolic plane |
| 5/1 | Hyperbolic polygons, hyperbolic structures on surfaces |
|
| 5/3 | Teichmüller space and moduli space |
|
| 6 | 5/6 | More on the moduli space of surfaces |
| 5/8 | More on the moduli space of surfaces |
|
| 5/10 | (G,X) structures on manifolds |
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| 7 | 5/13 | Examples of (G,X) structures: hyperbolic, spherical, Euclidean geometries |
| 5/15 | More hyperbolic geometry |
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| 5/17 | More hyperbolic geometry |
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| 8 | 5/20 | Prime decomposition of 3-manifolds |
| 5/22 | More on prime decompositions |
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| 5/24 | Incompressible surfaces |
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| 9 | 5/27 | Memorial Day Holiday (no class) |
| 5/29 | JSJ decompositions and related topics |
|
| 5/31 | Seifert fibered spaces, graph manifolds |
|
| 10 | 6/3 | Thurston's 8 geometries |
| 6/5 | Statement of the geometrization theorem |
|
| 6/7 | Computing hyperbolic structures in practice |
References
References for details and further reading.
- Week 1
- Farb and Margalit, A primer on mapping class groups, Chapters 1–3 (skim: their treatment is much more comprehensive than ours).
- Homotopy and isotopy agree for simple closed curves: Farb-Margalit Proposition 1.10 (using hyperbolic geometry)
- The smooth and topological mapping class groups agree for surfaces: see D. B. A. Epstein, "Curves on 2-manifolds and isotopies" for a modern explanation that topological agrees with PL. The original proof is due to R. Baer (in German), with follow-up work by W. Mangler and W Brodel, also in German. Going from there to smooth is, I think, not hard. The previous result is a key step in Baer's proof.
- Week 2
- Farb and Margalit, A primer on mapping class groups, Chapters 3 and 4 (selected portions, again).
- Lickorish, An Introduction to Knot Theory, Chapter 12 (through Corollary 12.8).
- Smale's Theorem: some online notes by and some more online notes by Alexander Kupers and some more online notes by Alexander Kupers.
- A key step in that proof is the Poincaré-Bendixson Theorem. See, for instance, Chapter 14 of Introduction to the Modern Theory of Dynamical Systems by Katok and Hasselblatt.
- I haven't bumped into a good modern reference for Macbeath's Theorem, but it's Theorem 1 in his paper "Groups of homeomorphisms of a simply connected space"
- Week 3
- Lickorish, An Introduction to Knot Theory, Chapter 12 (the rest).
- Saveliev, Lectures on the Topology of 3-Manifolds, Chapters 1 and 2.
- Gompf and Stipsicz, 4-manifolds and Kirby Calculus, Sections 4.1–4.3.
- Morse functions and handle decompositions: Milnor's books Morse Theory (which proves Morse functions give CW decompositions) and Lectures on the h-cobordism Theorem (which proves they give handle decompositions, talks about rearranging handles, and more).
- Thurston, Three-Dimensional Geometry and Topology, Section 3.10 talks about smoothing triangulated 3-manifolds.
- Week 4
- Saveliev, Lectures on the Topology of 3-Manifolds, Chapter 3. There are also some nice exercises here.
- Gompf and Stipsicz, 4-manifolds and Kirby Calculus, Sections 5.1–5.3. This also has extensive references and history, not to mention many exercises.
- Dale Rolfsen, Knots and Links, Chapter 8, especially Sections F, G, and H.
- Kirby and Scharlemann, “Eight faces of the Poincaré homology 3-sphere.”
- Week 5
- Farb and Margalit, A Primer on Mapping Class Groups, Chapter 1.
- Thurston, Three-Dimensional Geometry and Topology, Chapter 2 and Section 4.6.
- Thurston, The Geometry and Topology of Three-Manifolds, Section 5.3. (These are the famous, unpublished Thurston notes.)
- Purcell, Hyperbolic Knot Theory, Section 2.1. Available on the arXiv.
- Weeks 6 and 7
- Hatcher's Notes on Basic 3-Manifold Topology.
- Weeks 8–10
- Thurston, Three-Dimensional Geometry and Topology, Sections 3.7 and 3.8
- Scott, “The Geometries of 3-Manifolds”. Available online here.
- Purcell, Hyperbolic Knot Theory (see above).
- Aschenbrenner, Friedl, and Wilson, “Decision problems for 3-manifolds and their fundamental groups”.
- William Menasco, “Polyhedra representation of link complements”. In Low Dimensional Topology, Ed. by Samuel J. Lomonaco, Jr.. Contemporary Mathematics, Vol 20 (1983). (Available online through our library; follow the link above.)