Math 284: graduate seminar on three-manifolds

Course Details:

    Monday & Wednesday

    12:50-2:05

    381U


Office Hours by appointment

   380-382E



Three-manifolds are fundamental objects in geometric topology.  They’ve been studied for close to a century, yet they’re still central to contemporary research.



There’s no pretense that this is a systematic or complete introduction to three-manifolds.  Instead, we’ll use an interesting recent theorem of Andras Juhasz as an excuse to tour classical and contemporary topics.  Starting from scratch, we’ll try to end the course with the proof that sutured Floer homology detects knot genus.  Topics are selected to support this goal, so we’ll begin with topological tools (Heegaard diagrams, incompressible surfaces, sutured manifolds) and then move into Heegaard Floer theory in the second half of the course. 


Syllabus:

Week 1: prime and irreducible three-manifolds, Heegaard splittings, Heegaard diagrams

Week 2: Seifert surfaces, knot genus, surface complexity, Thurston norm

Week 3: sutured manifolds, tautness, hierarchies, disc decompositions

Week 4: applications of sutured manifolds, adapted Heegaard diagrams

Week 5: Sutured Floer Homology I

Week 6: Sutured Floer Homology II

Week 7: SFH of products and non-taut manifolds

Week 8: SFH and surface decompositions

Week 9: the loose ends and leftovers


References (through Week 2):

Marc Lackenby’s notes on 3-dimensional manifolds (scroll to the bottom)

Morse Theory, J. Milnor. Annals of Mathematics Studies, No. 51, Princeton University    

     Press (1963)

A norm for the homology of 3-manifolds, W. P. Thurston. Memoirs of the American

     Mathematical Society, Vol, 59, No. 336, American Mathematical Society (1986)