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.
More specifically;
Heegaard Diagrams
spin^c structures
homology
Poincaré Duality.
References include Ozsváth and Szabó's ``An introduction to Heegaard Floer homology" and Hom's "Lecture Notes on Heegaard Floer homology"
More specifically;
Gradings on Heegaard Floer homology.
Cobordism maps and exact triangles
d-invariants and the homology cobordism group
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".
More specifically;
Involutive d-invaraints, exact triangles
Naturality of Heegaard Floer homology
See Hendricks-Manolescu's paper; Involutive Heegaard Floer homology.
More specifically;
Knot Floer Homology.
Large surgery formula for Heegaard Floer homology
Mallick's large surgery formula for surgery on equivariant knots.
Dai-Kang-Mallick-Park-Stoffregen's result that the (2,1)-cable of the figure eight knot is not slice. Here is a Quanta article about this result.