Involutive Heegaard Floer homology learning seminar

Ali Naseri Sadr and I organised a learning seminar on involutive Heegaard Floer homology.

January 31st: Nathan Geist 

Prelude to Heegaard Floer Homology

This talk will provide the basic tools needed to construct HF homology. I'll start by discussing Heegaard diagrams and how they are a useful way of describing any closed oriented 3-manifold. In the next we'll see how to obtain Heegaard diagrams from Morse functions and their gradient vector fields. Finally, I'll give a quick overview of Spin^c-structures and their connection to second cohomology.

February 7th: Mira Wattal

Fighting my demons (Streptococcus Group A) with doubly pointed Heegaard diagrams and Spin^c structures

We will continue to build towards involutive floer homology by introducing new machinery such as doubly pointed Heegaard diagrams and Spin^C structures. Exercises will abound in this talk, but not streptococcus. (The author and this talk have been on antibiotics since Thursday.) 

February 14th: Kevin Yeh

Introduction to Heegaard Floer Homology

In this talk, I will give a quick introduction to Heegaard Floer Homology—a 3-manifold invariant due to Ozsváth and Szabó, constructed from a Heegaard diagram associated to a 3-manifold. The construction is inspired by the works of Floer and his celebrated Lagrangian Floer homology theory, which in turn takes inspiration from Morse homology. Apart from the construction, I will also mention some important properties of this invariant that will be important for future talks of this reading group; such as the splitting over Spin-C structures, as well as some important algebraic properties. If time permits I may mention the construction of knot Floer homology, a flavor of Heegaard Floer homology, which is in fact a knot invariant, built up from a doubly-pointed Heegaard diagram associated to a knot.

February 21st: Joe Boninger

Heegaard Floer homology has a TQFT!

We’ll look at how cobordisms between 3-manifolds induce chain maps in Heegaard Floer homology. Along the way, we’ll introduce the important concepts of Heegaard triples and holomorphic triangles.

February 28th: Nathan Geist

Is it $S^{3}$ or not!?!? An introduction to the Heegaard-Floer d-invariant

This week I'll define the d-invariant for rational homology spheres and give its basic properties. Some applications include the rational homology cobordism group, Donaldson's diagonalizability theorem, and surgeries giving lens spaces.

March 14th:  Ali Naseri Sadr

Definition of HFI

We will introduce the conjugation symmetry in Heegaard-Floer homology. Afterwards, we use this symmetry and first order naturality in HF to define HFI. We will finish by presenting some of HFI properties. 

March 21st:  Ali Naseri Sadr

Definition of HFI (part 2)

We will introduce the conjugation symmetry in Heegaard-Floer homology. Afterwards, we use this symmetry and first order naturality in HF to define HFI. We will finish by presenting some of HFI properties. 

March 28th: Kevin Yeh

Involutive Heegaard Floer Correction Terms

I will go over the analog of the Heegaard Floer correction term in the involutive setting. I will also go over some applications of these correction terms, in particular mention some use cases in which the ordinary, non-involutive correction term is not sufficient.

April 11th: Fraser Binns

An infinite rank summand of the homology concordance group

I will discuss various concordance and homology cobordism groups. A central question in the study of these groups is to determine their algebraic structure. I will discuss some techniques for approaching these questions, including some from involutive Heegaard Floer homology. I will also give an example of an infinite rank summand of a quotient of the integer homology concordance group. This example is from joint work with Hugo Zhou.

May 2nd: Fraser Binns

Large Surgery Formulae

Given a knot K, one can recover the Heegaard Floer homology of large integer surgeries on K from the knot Floer chain complex of K by work of Ozváth-Szabó and J.Rasmussen. In this talk I will discuss this and an analogous result in the involutive setting due to Hendricks-Manolescu.

May 9th: Fraser Binns

Large Surgery Formulae part 2

I will discuss the proof of Hendricks-Manolescu's large surgery formula in involutive Heegaard Floer homology.

May 16th: Ali Naseri Sadr

The 2,1 cable of figure eight is not slice.


The 2,1 cable of fig eight was a potential counterexample for the slice-ribbon conjecture. I will talk about a paper that shows this knot is not slice using Heegaard-Floer theory.

Topics

Heegaard Diagrams and 3-manifolds

More specifically;

References include  Ozsváth and Szabó's ``An introduction to Heegaard Floer homology" and Hom's "Lecture Notes on Heegaard Floer homology"

Heegaard Floer Homology

More specifically;

References include those mentioned above and Ozsváth-Szabó's original paper "Holomorphic disks and topological invariants for closed three-manifolds", "Absolutely Graded Floer homologies and intersection forms for four-manifolds with boundary".

Involutive Heegaard Floer Homology

More specifically;

See Hendricks-Manolescu's paper; Involutive Heegaard Floer homology. 

Applications of Involutive Heegaard Floer homology

More specifically;