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:

• Notes on basic 3-manifold topology, by Allen Hatcher
• An introduction to geometric topology, by Bruno Martelli
• The geometries of 3-manifolds, by Peter Scott

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