Research

Constructed by considering the moduli space of solutions to SU(2) gauge theory and ASD (anti-self-dual) equations on 3- and 4-manifolds, where the solutions are called instantons. Here, the ASD equation is a special case of the Yang–Mills equation. This homology is closely related to SU(2) representations of the fundamental group of the manifold.

Introductory references:

1. Floer Homology Group in Yang-Mills theory, Simon K. Donaldson
2. The Geometry of Four-Manifolds, Simon K. Donaldson, Peter B. Kronheimer
3. Lectures of Tomasz S. Mrowka, notes by Donghao Wang, related videos
4. The Floer Memorial Volume, edited by Helmut Hofer, Clifford H. Taubes, Alan Weinstein, Eduard Zehnder
5. Khonovanov homology is an unknot-detector, Peter B. Kronheimer, Tomasz S. Mrowka

Also known as Seiberg-Witten Floer homology, constructed by considering the solution space of Seiberg-Witten equations on 3- and 4-manifolds with spin^c structures. Here, monopoles refer to solutions of these equations. This homology is a U(1) gauge theory, which is geometrically easier to study due to its abelian (commutative) nature.

Introductory references:

1. Monopoles and Three-Manifolds, Peter B. Kronheimer, Tomasz S. Mrowka
2. The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds, John W. Morgan
3. Lectures on Seiberg-Witten Invariants, John Douglas Moore
4. Lecture videos of Andriy Haydys, course introduction
5. Monopoles and lens space surgeries, Peter B. Kronheimer, Tomasz S. Mrowka

Constructed via Heegaard diagrams and Lagrangian Floer homology on symmetric products of surfaces. This construction easily generalizes to knots, sutured 3-manifolds, 3-manifolds with boundary, and other cases, and can be computed combinatorially by choosing nice Heegaard diagrams. This homology was proven to be isomorphic to monopole Floer homology by Kutluhan–Lee–Taubes. Another proof by Colin–Ghiggini–Honda and Taubes shows that both Floer homologies are isomorphic to embedded contact homology (ECH, see below). However, the maps corresponding to 4-dimensional cobordisms have not been proven to be the same under these isomorphisms.

Introductory references:

1. References in Peter S. Ozsváth’s homepage
2. Lecture notes on Heegaard Floer homology, Jennifer Hom, related videos
3. Holomorphic disks and topological invariants for closed three-manifolds, Peter S. Ozsváth, Zoltán Szabó
4. Holomorphic disks and three-manifolds invariants: Properties and applications, Peter S. Ozsváth, Zoltán Szabó

The above three Floer homologies can be generalized to knots, sutured 3-manifolds, 3-manifolds with boundary, and other cases. Among them, Heegaard Floer homology is easier to generalize, while instanton Floer homology and monopole Floer homology are more indirect but easier for studying the naturality of cobordism maps. In particular, the knot versions of all three Floer homologies are bigraded homology groups, and their graded Euler characteristics are all the Alexander polynomial. Kronheimer–Mrowka conjecture that the knot and sutured manifold versions of the three Floer homologies are isomorphic over complex coefficients, and the isomorphism between monopole Floer homology and Heegaard Floer homology has been proven by Lekili and Baldwin–Sivek. The following references cover various types of Floer homology generalizations and can be selected for reading.

Introductory references:

1. An introduction to knot Floer homology, Ciprian Manolescu
2. Grid Homology for Knots and Links, Peter S. Ozsváth, András I. Stipsicz, Zoltán Szabó
3. Holomorphic discs and sutured manifolds, András Juhász
4. Knot Floer homology detects fibred knots, Yi Ni
5. Knots, sutures, and excision, Peter B. Kronheimer, Tomasz S. Mrowka
6. Bordered Heegaard Floer homology, Robert Lipshitz, Peter S. Ozsváth, Dylan P. Thurston

In addition to generalizations to different 3-manifold cases, Floer homology can be extended by incorporating additional structures (such as spectra, equivariant, involutive, real) to obtain finer topological invariants. Different Floer homologies, due to their construction, are suited to different generalization constructions and topological applications. The following references cover various types of Floer homology generalizations and can be selected for reading.

Introductory references:

1. Lectures on the triangulation conjecture, Ciprian Manolescu
2. On the intersection forms of spin four-manifolds with boundary, Ciprian Manolescu
3. Lectures on monopole Floer homology, Francesco Lin
4. Equivariant aspects of singular instanton Floer homology, Aliakbar Daemi, Christopher Scaduto
5. Involutive Heegaard Floer homology, Kristen Hendricks, Ciprian Manolescu
6. Real Heegaard Floer homology, Gary Guth, Ciprian Manolescu
7. Monopole Floer homology and real structures, Jiakai Li

Contact structures are important geometric structures on 3-manifolds, closely related to symplectic geometry and low-dimensional topology. Through 3-manifold Floer homology theory, invariants can be constructed for contact structures. These invariants play an important role in the classification and study of contact structures, especially for tight contact structures and Stein-fillable contact structures.

Introductory references:

1. An Introduction to Contact Topology, Hansjörg Geiges
2. On the classification of tight contact structures I, Ko Honda
3. Heegaard Floer homologies and contact structures, Peter Ozsváth, Zoltán Szabó
4. The contact invariant in sutured Floer homology, Ko Honda, William H. Kazez, Gordana Matić
5. Monopoles and contact structures, Peter B. Kronheimer, Tomasz S. Mrowka
6. A contact invariant in sutured monopole homology, John A. Baldwin, Steven Sivek
7. Instanton Floer homology and contact structures, John A. Baldwin, Steven Sivek

Constructed by considering periodic orbits of Hamiltonian systems on symplectic manifolds, with generators being critical points of action functionals on loop spaces. This homology is closely related to the Arnold conjecture.

Introductory references:

1. Morse Theory and Floer Homology, Michèle Audin, Mihai Damian
2. J-holomorphic Curves and Symplectic Topology, Dusa McDuff, Dietmar Salamon

Also known as Lagrangian intersection Floer homology, constructed by considering intersections of Lagrangian submanifolds and pseudo-holomorphic disks. This homology theory is the foundation of Fukaya categories and plays an important role in mirror symmetry and low-dimensional topology.

Introductory references:

1. A beginner’s introduction to Fukaya categories, Denis Auroux
2. Fukaya Categories and Picard-Lefschetz Theory, Paul Seidel
3. Lectures of Paul Seidel, notes by Donghao Wang

Constructed by considering embedded Reeb orbits on 3-dimensional contact manifolds. This homology was proven to be isomorphic to Heegaard Floer homology and monopole Floer homology by Colin–Ghiggini–Honda and Taubes, allowing results in contact topology and 3-dimensional topology to be translated between each other.

Introductory references:

1. Lecture notes on embedded contact homology, Michael Hutchings
2. Embedded contact homology and its applications, Michael Hutchings

Constructed by considering cube resolutions of links and Frobenius algebras, yielding bigraded homology groups whose graded Euler characteristic equals the Jones polynomial. From the spectral sequence from Khovanov homology to Lee homology, one can obtain the Rasmussen invariant, which provides a combinatorially computable lower bound for the slice genus of knots.

Introductory references:

1. A categorification of the Jones polynomial, Mikhail Khovanov
2. Khovanov homology and the slice genus, Jacob Rasmussen

Also known as sl(n) homology, where the n=2 case corresponds to Khovanov homology. Constructed by considering representations of sl(n) quantum groups and cube resolutions of links, it provides finer algebraic invariants for links. This homology theory is constructed through the method of matrix factorization, and its graded Euler characteristic equals the sl(n) polynomial, which is a special case of the HOMFLY-PT polynomial. Furthermore, there exists a spectral sequence from HOMFLY-PT homology (see below) to Khovanov–Rozansky homology.

Introductory references:

1. Matrix factorizations and link homology, Mikhail Khovanov, Lev Rozansky
2. Matrix factorizations and link homology II, Mikhail Khovanov, Lev Rozansky
3. Some differentials on Khovanov–Rozansky homology, Jacob Rasmussen

Also known as HOMFLY-PT homology, it is a categorification of the HOMFLY-PT polynomial. Constructed by considering more general quantum group representations, it unifies Khovanov homology and Khovanov–Rozansky homology, providing a triply-graded homology theory for links, whose graded Euler characteristic equals the HOMFLY-PT polynomial. This homology theory has profound applications in algebraic geometry and representation theory, and is closely related to the Hochschild homology of Soergel bimodules.

Introductory references:

1. Triply-graded link homology and Hochschild homology of Soergel bimodules, Eugene Gorsky, Alexei Oblomkov, Jacob Rasmussen, Vivek Shende
2. Hilbert schemes and y-ification of Khovanov–Rozansky homology, Eugene Gorsky, Matthew Hogancamp