Related papers: Lectures on monopole Floer homology

We recently defined an invariant of contact manifolds with convex boundary in Kronheimer and Mrowka's sutured monopole Floer homology theory. Here, we prove that there is an isomorphism between sutured monopole Floer homology and sutured…

We use closed geodesics to construct and compute Bott-type Morse homology groups for the energy functional on the loop space of flat $n$-dimensional tori, $n\ge 1$, and Bott-type Floer cohomology groups for their cotangent bundles equipped…

We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes…

Morse theory relates algebraic topology invariants and the dynamics of the gradient flow of a Morse function, allowing to derive information about one out of the other. In the case of the homology, the construction extends to much more…

We show that monopole Floer homology (as defined by Kronheimer and Mrowka) is isomorphic to the S^1-equivariant homology of the Seiberg-Witten Floer spectrum constructed by the second author.

In bounding the homology of a manifold, Forman's Discrete Morse theory recovers the full precision of classical Morse theory: Given a PL triangulation of a manifold that admits a Morse function with c_i critical points of index i, we show…

We define combinatorial Floer homology of a transverse pair of noncontractibe nonisotopic embedded loops in an oriented 2-manifold without boundary, prove that it is invariant under isotopy, and prove that it is isomorphic to the original…

In this remark, we show how the monopole Fr{\o}yshov invariant, as well as the analogues of the Involutive Heegaard Floer correction terms $\underline{d},\overline{d}$, are related to the $\mathrm{Pin}(2)$-equivariant Floer homology…

We construct an algebraic version of Lagrangian Floer homology for immersed curves inside the pillowcase. We first associate to the pillowcase an algebra A. Then to an immersed curve L inside the pillowcase we associate an A infinity module…

We prove that the rank of the cohomology of a closed symplectic manifold with coefficients in a field of characteristic $p$ is smaller than the number of periodic orbits of any non-degenerate Hamiltonian flow. Following Floer, the proof…

We define Floer homology for a time-independent, or autonomous Hamiltonian on a symplectic manifold with contact type boundary, under the assumption that its 1-periodic orbits are transversally nondegenerate. Our construction is based on…

This is an expansion on my talk at the Geometry and Topology conference at McMaster University, May 2004. We outline a program to relate the Heegaard Floer homologies of Ozsvath-Szabo, and Seiberg-Witten-Floer homologies as defined by…

Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian flows on solid tori, periodic flow-lines of which define braid (conjugacy) classes, up to full twists. We examine the dynamics…

Floer field theory is a construction principle for e.g. 3-manifold invariants via decomposition in a bordism category and a functor to the symplectic category, and is conjectured to have natural 4-dimensional extensions. This survey…

This article arose from a series of three lectures given at the Banach Center, Warsaw, during period of 24 March to 13 April, 2003. Morse functions are useful tool in revealing the geometric formation of its domain manifolds $M$. They…

This is a companion paper to earlier work of the authors, which interprets the Heegaard Floer homology for a manifold with torus boundary in terms of immersed curves in a punctured torus. We prove a variety of properties of this invariant,…

In this paper, Floer homology for Lagrangian submanifolds in an open symplectic manifold given as the complement of a smooth divisor is discussed. The main new feature of this construction is that we do not make any assumption on positivity…

We review the construction of Heegaard Floer homology for closed three-manifolds and also for knots and links in the three-sphere. We also discuss three applications of this invariant to knot theory: studying the Thurston norm of a link…

It is well-known that the reduced Floer homology of a rational homology sphere admitting a taut foliation does not vanish. We strengthen this by showing that (when thought of as an $\mathbb{F}[U]$-module) it also admits a direct…

This is a survey of bordered Heegaard Floer homology, an extension of the Heegaard Floer invariant HF-hat to 3-manifolds with boundary. Emphasis is placed on how bordered Heegaard Floer homology can be used for computations.