# Three-manifolds, Spring 2016

This course is an introduction to some topics in classical three-manifold topology, including
• Prime decomposition.
• Torus / JSJ decomposition.
• The loop and sphere theorems.
• Sutured manifold hierarchies.
We'll also probably talk about some hyperbolic geometry, and discuss geometrization.

## References

• A. Hatcher, Notes on Basic 3-Manifold Topology. This reference covers the prime and torus decompositions and the loop and sphere theorems. This is the main reference for the first part of the course.
• W. Thurston, Three-dimensional geometry and topology. Among many other things, this book explains how to go back and forth between smooth structures and triangulations in dimension three and lower.
• E. Moise, Geometric topology in dimensions 2 and 3. This book explains why three-manifolds have triangulations.
• Scott & Wall, Topological methods in group theory, chapter 5 of this book. This source explains Stallings Ends Theorem, as well as the proof we give of the Sphere Theorem.
• W. Thurston, A norm for the homology of 3-manifolds, is the second part of this book.
• More references can be found in the notes below.

## Notes:

(Date shown is last file update, not the date of the lecture.)
• Lecture 1, introduction to the course. (2016/02/03)
• Lecture 2, some smooth topology, Alexander's Theorem. (2016/02/08)
• Lecture 3, starting prime decomposition. (2016/02/07)
• Lecture 4, existence of prime decomposition. (2016/02/10)
• Lecture 5, uniqueness of prime decomposition, definition of incompressible surfaces. (2016/02/12)
• Lecture 6, incompressible surfaces and normal surfaces. (2016/02/28)
• Lecture 7, Haken finiteness, Seifert fibered spaces. (2016/02/28)
• Lecture 8, essential surfaces. (2016/05/06)
• Lecture 9, essential surfaces in SFS are vertical or horizontal. (2016/04/18)
• Lecture 10, orbifolds and surfaces in SFS. (2016/04/18)
• Lecture 11, uniqueness of torus decompositions. (2016/04/18)
• Lecture 12, uniqueness of torus decompositions, ctd. (2016/03/14)
• Lecture 13, uniqueness of torus decompositions, finished. Statement of Loop theorem and Dehn's Lemma. (2016/03/21)
• Lecture 14, Loop Theorem: building the tower and finding a disk. (2016/03/21)
• Lecture 15, finished proof of Loop Theorem. Representing 2-dimensional homology classes by nice surfaces. (2016/03/24)
• Lecture 16, applications of loop and sphere theorems. (2016/05/06)
• Lecture 17, reduction of sphere theorem to compact manifold with incompressible boundary. (2016/05/06)
• Lecture 18, outline of sphere theorem from Stallings ends theorem. (2016/04/06)
• Lecture 19, completing sketch of sphere theorem and Stallings ends theorem. Said a few words about what's next. (2016/04/27)
• Lecture 20, fibering, Thurston norm. (2016/04/28)
• Lecture 21, Thurston norm is a seminorm. (2016/04/18)
• Lecture 22, Integral forms have polyhedral unit norm ball. Tischler's theorem. (2016/05/04)
• Lecture 23, Fibers are norm minimizing. Euler class computes norm in a neighborhood of a fibered class. (2016/05/02)
• Lecture 24. Decomposition into Thurston cones. If a class is fibered, then it lies in an open cone on a top-dimensional face, and every other class in the cone is fibered. (2016/05/02)
• Lecture 25. Haken manifolds have hierarchies. Length of a Haken manifold is at most thrice the closed Haken number. (2016/05/22)
• Lecture 26. Sutured manifolds definitions. (2016/05/22)
• Lecture 27. Decomposition surfaces, guts, windows. (2016/05/16)
• Lecture 28. RFRS for hyperbolic 3-manifolds.