MAT 566: Topics in Differential Topology

Fukaya Categories and Floer homology of 3-manifolds

Spring 2021

Time: TTh 9:30-10:50am. Lectures held via Zoom (email me for zoom link).
Office hours: By appointment.

Course Description: We will introduce the Fukaya category of a symplectic manifold, with a particular focus on the case of surfaces (where many of definitions can be made combinatorial). In this case the objects of the Fukaya category are (decorated) immersed curves in the surface and morphisms count pseudoholomorphic disks. We will also explore the role of immersed curves in Heegaard Floer homology, particularly in the context of knot Floer homology and bordered Floer homology, and also related invariants such as Khovanov homology. We will highlight some recent applications of the interplay between immersed curves in surfaces and these 3-dimensional invariants.

References

Hanselman, Rasmussen, Watson, Bordered Floer homology for manifolds with torus boundary via immersed curves
            Introduces immersed curves to describe bordered Floer homology.
Kotelskiy, Watson, Zibrowius, Immersed curves in Khovanov homology
            Applies immersed curves to the Khovanov homology of a 4-ended tangle.

Tentative schedule

Date	Topics	Lecture notes
2/2	Overview: Fukaya categories and immersed curves	Lecture 1
2/4	A∞-categories	Lecture 2
2/9	Symplectic background, Lagrangian Floer cohomology	Lecture 3
2/11	Lagrangian Floer cohomology, continued	Lecture 4
2/16	Invariance, product structure	Lecture 5
2/18	A∞-structure	Lecture 6
2/23	Finish A-infinity relation, some examples	Lecture 7
2/25	Combinatorial Floer homology in surfaces	Lecture 8
3/2	Combinatorial Fukaya category	Lecture 9
3/4	Twisted complexes	Lecture 10
3/9, 3/11	Twisted complexes and generation	Lecture 11
3/11	Generators, local systems, Tw(Fuk(T^2))	Lecture 12
	Spring Break
3/23	Floer homology of immersed train tracks	Lecture 13
3/25	Quasi-isomorphic train tracks	Lecture 14
3/30	Classifying train tracks in punctured torus	Lecture 15
4/1	Classification in punctured torus, continued	Lecture 16
4/6	Uniqueness, partially wrapped Fukaya category	Lecture 17
4/8	Classification in partially wrapped setting	Lecture 18
4/13	Type D structures, Heegaard Floer homology	Lecture 19
4/15	Bordered Floer invariants as immersed curves	Lecture 20
4/20	Pairing theorem, some gradings	Lecture 21
4/20	L-space gluing result	Lecture 22
4/27	L-space gluing result, continued	Lecture 23
4/29	Immersed curves and knot Floer homology	Lecture 24

Homework