Related papers: Floer Homology: From Generalized Morse-Smale Dynam…

Based on the concept of manifold valued generalized functions we initiate a study of nonlinear ordinary differential equations with singular (in particular: distributional) right hand sides in a global setting. After establishing several…

We construct Morse homology groups associated with any regular function on a smooth complex algebraic variety, allowing singular and non-compact critical loci. These groups are generated by critical points of a certain large pertubation of…

We construct Morse-Smale-Witten complex for an effective orientable orbifold. For a global quotient orbifold, we also construct a Morse-Bott complex. We show that certain type of critical points of a Morse function has to be discarded to…

We apply the methods of Heegaard Floer homology to identify topological properties of complex curves in the complex projective plane. As one application, we resolve an open conjecture that constrains the Alexander polynomial of the link of…

The periodic Floer homology of a surface symplectomorphism, defined by the first author and M. Thaddeus, is the homology of a chain complex which is generated by certain unions of periodic orbits, and whose differential counts certain…

We generalize the Cohen-Jones-Segal construction to the Morse-Bott setting. In other words, we define framings for Morse-Bott analogues of flow categories and associate a stable homotopy type to this data. We use this to recover the stable…

Floer theory relates the dynamics of Hamiltonian isotopies and the homology of the ambient manifold. It was extended to similarly relate the dynamics of symplectic isotopies and the Novikov homology associated to their flux. We discuss this…

In this paper we describe and continue the study begun by the author, Jones, and Segal, of the homotopy theory that underlies Floer theory. In that paper the authors addressed the question of realizing a Floer complex as the celluar chain…

For an adiscal or monotone regular coisotropic submanifold $N$ of a symplectic manifold I define its Floer homology to be the Floer homology of a certain Lagrangian embedding of $N$. Given a Hamiltonian isotopy $\phi=(\phi^t)$ and a…

Computing homology and cohomology is at the heart of many recent works and a key issue for topological data analysis. Among homological objects, homology generators are useful to locate or understand holes (especially for geometric…

We construct a filtration by ideals on quantum cohomology for symplectic manifolds with a Hamiltonian $S^1$-action that extends to a pseudoholomorphic $\mathbb{C}^*$-action. These spaces include all Conical Symplectic Resolutions, in…

In~\cite{rotvandervorst} a homology theory --Morse-Conley-Floer homology-- for isolated invariant sets of arbitrary flows on finite dimensional manifolds is developed. In this paper we investigate functoriality and duality of this homology…

We assign to each nondegenerate Hamiltonian on a closed symplectic manifold a Floer-theoretic quantity called its "boundary depth," and establish basic results about how the boundary depths of different Hamiltonians are related. As…

This paper extends the definition of Rabinowitz Floer homology to non-compact hypersurfaces. We present a general framework for the construction of Rabinowitz Floer homology in the non-compact setting under suitable compactness assumptions…

We construct topological invariants, called abstract weak orbit spaces, of flows and homeomorphisms on topological spaces, to describe both gradient dynamics and recurrent dynamics. In particular, the abstract weak orbit spaces of flows on…

We show that if K: P \to R is an autonomous Hamiltonian on a symplectic manifold (P,\Omega) which attains 0 as a Morse-Bott nondegenerate minimum along a symplectic submanifold M, and if c_1(TP)|_M vanishes in real cohomology, then the…

We use Liu-Tian's virtual moduli cycle methods to construct detailedly the explicit isomorphism between Floer homology and quantum homology for any closed symplectic manifold that was first outlined by Piunikhin, Salamon and Schwarz for the…

Robin Forman's highly influential 2002 paper A User's Guide to Discrete Morse Theory presents an overview of the subject in a very readable manner. As a proof of concept, the author determines the topology (homotopy type) of the abstract…

We construct an action selector on aspherical symplectic manifolds that are closed or convex. Such selectors have been constructed by Matthias Schwarz using Floer homology. The construction we present here is simpler and uses only Gromov…

We introduce a simple combinatorial method for computing all versions of the knot Floer homology of the preimage of a two-bridge knot K(p,q) inside its double-branched cover, -L(p,q). The 4-pointed genus 1 Heegaard diagram we obtain looks…