## Math 9500: Introduction to 3-Manifolds

### Fall 2022

 Meets: Tue/Thu 11:00-12:20 in Wachman Hall, room 617 Instructor: David Futer Office: 1026 Wachman Hall Office Hours: Tue 1:30-3:00, Wed 10:30-12:00, or by appointment E-mail: dfuter at temple.edu Phone: (215) 204-7854

Course content: This course is motivated by the homeomorphism problem. Given a pair of n-manifolds M and N, is there an algorithmic procedure to decide whether they are homeomorphic? This problem is trivial in dimension 1, fairly easy in dimension 2, and impossible in all dimensions n ≥ 4. Dimension n=3 is the middle ground, where the problem is solvable but fairly hard.

The solution to the homeomorphism problem in dimension 3 relies on the Geometrization theorem, posed as a conjecture by Thurston around 1980 and proved by Perelman in 2003. This result says that every 3-manifold can be canonically subdivided into pieces, such that every piece has one of 8 homogeneous geometries. So we will study how exactly the cutting procedure works, what the 8 geometries are, and how they can be used to address the homeomorphism problem.

Surfaces embedded in 3-manifolds will play a starring role. We will study normal surfaces, incompressible surfaces, and Haken hierarchies. We will also discuss constructions of 3-manifolds from mapping classes of surfaces, via Heegaard splittings and fibrations. Time permitting, we will discuss some "virtual problems" about covers of 3-manifolds.

References:

Prerequisites: Math 8061-62.

Grading: Grades will be assigned based on homework and a presentation toward the end of the semester.

### Detailed schedule

This will be gradually filled in as the semester progresses.
8/23 The homeomorphism problem; surfaces Conway's ZIP proof
8/25 Geometric examples, lens spaces Martelli, p. 303-310 Play with Curved spaces app
8/30 I-bundles, connected sums Martelli, p. 270-274
9/1 Morse theory, Alexander's theorem Martelli, p. 274-276Homework 1, due 9/13
9/6 Prime vs irreducible manifolds Martelli, p. 276-278
9/8 Normal surface theory Martelli, p. 279-282
9/13Prime decomposition Martelli, p. 282-284
9/15Incompressible surfaces Martelli, p. 287-289
9/20 Essential surfaces Martelli, p. 285; 290-291
9/22 Haken Manifolds Martelli, p. 294-296
9/27 Haken hierarchies Martelli p. 293, 296-297; Hatcher Lemma 3.5
9/29 The Loop/Disk Theorem Martelli, p. 298-300; Hatcher Theorem 3.1; Kent notes Sec. 8
10/4 Circle bundles over surfaces with boundary Martelli, p. 310-312.
10/6 Circle bundles over closed surfaces Martelli, p. 312-313
10/11 Seifert fibrations Martelli, p. 314-318 Homework 2, due 10/20
10/13 Classification of SFS, Torus decomposition Martelli, Section 10.4; Hatcher, Ex. on page 13
10/18 Torus (JSJ) Decomposition Martelli, p. 364-366
10/20 Geometric structures on surfaces Martelli, p. 160-162
10/25 Three-dimensional geometries: S3 Martelli, p. 370-379; Scott, p. 450-457 Choose a presentation topic by next week
10/27 Euclidean and S2xR geometries Martelli, p. 380-384
11/1 H2xR geometry, start of Nil Martelli, p. 384-388
11/3 Nil geometry Martelli, p. 388-392
11/8 SL2R geometry Martelli, p. 392-396; Scott, p. 462-467
11/10 Sol geometry Martelli, p. 396-399
11/15 H3 geometry Martelli, p. 62-68
11/17 Commuting isometries of H3 Martelli, p. 115-116
11/29 Geometrization Martelli, p. 400-403
11/30 Presentations: Rob, Ross
12/1 Presentations: Andrew, Brandis, Dipika

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