Math 235+236: Heegaard Floer homology

Winter+Spring 2025

Instructor: Sucharit Sarkar.
Class: MWF 12-12:50, MS 6221.
Office hrs: By appointment.

Exams and grading: Since this is a graduate topics course, there is no grading per se. However, for the undergraduate students attending the course and who will need a grade, their grade will be entirely based on attendance and HW which is due at the end of the quarter. The HW is the PDF below (which will be updated after every lecture) and involves working out the details that I skipped in class. There will be no finals.
HW

Topics: Heegaard Floer homology. This is a fairly recent and extremely active field of research in low-dimensional topology. It associates new homology theories to 3-manifolds and knots and links inside them, and these invariants carry a lot of geometric information about the underlying objects; the theory has been extended for 2-manifolds and 4-manifolds as well. Being recent, there are no good books on the topic. So we will cover (a subset of) the material from the following sources (arranged chronologically).

Lecture notes:

• An introduction to Heegaard Floer homology (Ozsvath-Szabo)
• Lectures on Heegaard Floer homology (Ozsvath-Szabo)
• An introduction to knot Floer homology (Manolescu)
• Lecture notes on Heegaard Floer homology (Manolescu)
• Heegaard Floer homologies: lecture notes (Lipshitz)
• Lecture notes on Heegaard Floer homology (Hom)

Original Ozsvath-Szabo papers:

• Holomorphic disks and topological invariants for closed three-manifolds
• Holomorphic disks and three-manifold invariants: properties and applications
• Holomorphic triangles and invariants for smooth four-manifolds
• Holomorphic disks and knot invariants