## Applications of Heegaard Floer Homology

Heegaard Floer theory was introduced in 2001 by Ozsvath and Szabo. The theory provides a package of invariants for 3-manifolds, smooth 4-manifolds and knots. It has proved to be a powerful tool, leading to many important advances in low-dimensional topology and solutions to long-standing problems that resisted previously known methods. Glorious as it sounds, it comes with a price; it is difficult and technical. Hence, this seminar aims to study some of its applications in order to understand it a little better. Our focus is on questions about knots and surgeries.

#### Resources

__Papers__

Eftekhary, "Heegaard Floer homology of pretzel knots" (2003)

Ghiggini, "Knot Floer homology detects genus-one fibred knots" (2006)

Hendricks and Manolescu, "Involutive Heegaard Floer homology" (2016)

Livingston, "Notes on the knot concordance invariant Upsilon" (2015)

Ni, "Knot Floer homology detects fibred knots" (2006)

Ozsváth and Szabó, "Holomorphic disks and genus bounds" (2003)

Ozsváth and Szabó, "Holomorphic disks and lens space surgeries" (2005)

__Videos__

Jennifer Hom, PCMI Talk Series on Heegaard Floer Homology

Yi Ni, Perspectives on Dehn Surgery Talk Series on Heegaard Floer Homology (go to the end of the page)

__Notes and Exercises__

Danny Calegari, Chapter on Foliations and Floer Theories

Jennifer Hom, Notes on Heegaard Floer Homology

Ciprian Manolescu, "Lectures on the triangulation conjecture"

Yi Ni, Exercises on Heegaard Floer Homology

George Torres, Notes on Contact Topology(from Bob Gompf's class)

__Books__

Yakov Eliashberg and William Thurston, *Confoliations*

Ieke Moerdijk and Janez Mrčun, *Introduction to Foliations and Lie Groupoids*

#### Schedule

This seminar meets in RLM 12.166 at 2:00pm.
Here's the tentative schedule for this seminar.:

__September 6/13:__ Introduction to HF & HFK (Kai)

*Resource: Jen Hom PCMI videos Lecture 2&3*

__September 20:__ HF & Surgery (Shiyu)

*Resource: Jen Hom PCMI videos Lecture 4*

*Yi Ni PoD videos is also a good source*

__September 27/October 4:__ Introduction to Taut Foliations (Cass)

*Resource: Introduction to Foliations and Lie Groupoids; Confoliations*

*Also, Calegari's chapter on Foliations and Floer Theory and George's Contact Topology notes*

__October 11 (at noon):__ Pizza Seminar - Knot Concordances and HF (Ceren)

__October 14:__ Jen is here! Go to Senior Topology and Colloquium seminars, and then dinner!

__October 18/25:__ HF and 3-genus (Hannah)

*Resource: Ozsváth and Szabó, Holomorphic disks and genus bounds*

__November 1:__ HF and Lens Space Surgeries (Jonathan)

*Resource: Ozsváth and Szabó, Holomorphic disks and lens space surgeries*

__November 8:__ Triangulation Conjecture Seminar - Involutive Heegaard Floer Homology, part 1 (Jonathan, 11:00 on 8th floor)

*Resource: Manolescu, Lectures on the triangulation conjecture*

__November 15:__ Triangulation Conjecture Seminar - Involutive Heegaard Floer Homology, part 2 (Kai, 11:00 on 8th floor)

*Resource: Manolescu, Lectures on the triangulation conjecture*

__November 22:__ Upsilon invariant (Ceren)

*Resource: Livingston, Notes on the knot concordance invariant Upsilon*