Jennifer Hom 2.1, Heegaard Floer homology (IAS | PCMI)
Channel: IAS | Park City Mathematics Institute | PCMI
Date: 10/07/2019

Heegaard Floer homology, defined by Ozsv ́ath-Szabo [OS04d], and knot Floer homology, defined by Ozsv ́ath-Szab ́o [OS04b] and independently Rasmussen [Ras03], are invariants of 3-manifolds and knots inside of them. These notes aim to provide an overview of these invariants and the relationship between them. 
We will describe our manifolds and knots via Heegaard diagrams, which we define in Section 1. From such diagrams, we will construct chain complexes for 3-manifolds, respectively knots, in Section 2, respectively Section 3, whose chain homotopy type (and in particular, homology) is independent of the choice of Heegaard diagram. Moreover, from the knot invariant associated to a knot K in S3, one can compute the 3-manifold invariant for any Dehn surgery along K; we discuss this relationship in the case of integral surgery in Section 4. 

This 4-lecture course is being presented at the 29th annual PCMI Summer Session taking place July 30–July 19 2019, at the Prospector Conference Center in Park City, Utah. The residential, three-week Summer Session is the flagship activity of the IAS | Park City Mathematics Institute (PCMI), an outreach program of the Institute for Advanced Study. The 2019 research theme is Quantum Field Theory and Manifold Invariants.

The PCMI Summer Session is an intensive program that includes several parallel sets of activities aimed at different groups of participants across the entire mathematics community. These activities include a program for mathematics researchers, eight mini-courses for graduate students on topics related to Quantum Field Theory and Manifold Invariants, two lecture series for undergraduate students, a program for faculty from predominantly undergraduate institutions, a workshop on equity and mathematics education for undergraduate faculty, and a professional development program for K–12 school teachers. All of PCMI’s groups meet simultaneously, pursuing individual courses of study and a meaningful amount of interaction both scientifically and socially.

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