arXiv research

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 theory. As a preliminary we investigate functoriality in Morse homology. Functor…

For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for i…

The paper constructs Morse complexes for orbifolds and shows their homologies are orbifold invariants.

An explicit isomorphism between Morse homology and singular homology is constructed via the technique of pseudo-cycles. Given a Morse cycle as a formal sum of critical points of a Morse function, the unstable manifolds for the negative gradient flow are compactified in a suitable way, such that gluing them appropriatel…

New Morse theory for path homology with coefficients.

The paper develops methods for calculating equivariant homology from Morse functions.

In this paper we study Morse homology and cohomology with local coefficients, i.e. "twisted" Morse homology and cohomology, on closed finite dimensional smooth manifolds. We prove a Morse theoretic version of Eilenberg's Theorem, and we prove isomorphisms between twisted Morse homology, Steenrod's CW-homology with loca…

New proof for discrete Morse theory using combinatorial construction.

Study homological mirror symmetry for Hirzebruch surfaces using Morse homotopy.

Study Morse complexity of manifolds and homology classes, proving bounds and implications.

Defines and calculates foliation homology from flows.

New pairing defined from Morse complexes for compact manifolds.

Homological mirror symmetry for toric Fano surfaces using Morse homotopy.

We give a new proof of the Morse Homology Theorem by constructing a chain complex associated to a Morse-Bott-Smale function that reduces to the Morse-Smale-Witten chain complex when the function is Morse-Smale and to the chain complex of smooth singular $N$-cube chains when the function is constant. We show that the ho…

In this paper, we define a relative Morse complex for manifold with boundary using the handlebody decomposition of the manifold. We prove that the homology of the relative Morse complex is isomorphic to the relative singular homology. Furthermore, we construct $A_\infty$-category structure on the relative Morse complex…

Constructs Morse homology for complex algebraic varieties.

Unified Morse-Bott-Smale chain complex, resolves well-definedness issue.

Extends Morse-Novikov Homology to include differential graded coefficients and fibration structures.

Two non-Morse-Bott Chern-Simons functions on homology 3-spheres.

Study rational homology of moduli space via Morse functions, proving stability phenomena.

An introduction to circle valued Morse theory and Novikov homology, from an algebraic point of view.

Notes on Morse Homology, focusing on gradient flow lines and semi-infinite dimensional cases.

Develops Morse homology with DG coefficients for manifolds and spaces.

New method connects curvature and Persistent Homology for networks.

The paper constructs Morse homology for functionals involving the p-Laplacian in Banach spaces.

Develops a method to compute Morse homology for clean but not necessarily transverse intersections.

Constructs flows of tori in sphere perturbations for Morse homology.

Discrete Morse theory simplifies Khovanov homology calculations.

New connection found between shape reconstruction methods and persistent homology.

Khovanov homology fails to differentiate certain slice disks.

Given a compact smooth manifold $M$ with non-empty boundary and a Morse function, a pseudo-gradient Morse-Smale vector field adapted to the boundary allows one to build a Morse complex whose homology is isomorphic to the (absolute or relative to the boundary) homology of $M$ with integer coefficients. Our approach simp…

The Morse-Novikov number MN(L) of a smooth link L in the three-dimensional sphere is by definition the minimal possible number of critical points of a regular circle-valued Morse function on the link complement (the term regular means that the Morse function must have nice behaviour in a tubular neighbourhood of L). No…

Transport functions for principal bundles and Morse homology with differential graded coefficients

The paper develops Morse homology for a class of elliptic partial differential equations.

We study Morse theory on noncompact manifolds equipped with exhaustions by compact pieces, defining the Morse homology of a pair which consists of the manifold and related geometric/homotopy data. We construct a collection of Morse data parametrized by cubes of arbitrary dimensions. From this collection, we obtain a fa…

Let $f:M \rightarrow \mathbb{R}$ be a Morse-Bott function on a finite dimensional closed smooth manifold $M$. Choosing an appropriate Riemannian metric on $M$ and Morse-Smale functions $f_j:C_j \rightarrow \mathbb{R}$ on the critical submanifolds $C_j$, one can construct a Morse chain complex whose boundary operator is…

Our objective is to develop a stratified Morse theory with tangential conditions. We define a continuous strata-wise smooth Morse function on an abstract stratified space by using control conditions and radiality assumptions on the gradient vector field. For critical points of a Morse function one can show that the loc…

We show how to construct homology bases for certain CW complexes in terms of discrete Morse theory and cellular homology. We apply this technique to study certain subcomplexes of the half cube polytope studied in previous works. This involves constructing explicit complete acyclic Morse matchings on the face lattice of…

Paper shows equivalence between MM and PH for n-D Morse functions.

Floer constructs homology from flow lines in generalized dynamical systems and combinatorial vector fields.

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 define the handle decompositions of $M$ from which the additive homologies $H_{\ast}(…

Study Morse models for torus algebra related to knot homology.

Simplifies fixing Khovanov homology functoriality.

Examples of Morse functions with integrable gradient flows on some classical Riemannian manifolds are considered. In particular, we show that a generic height function on the symmetric embeddings of classical Lie groups and certain symmetric spaces is a perfect Morse function, i.e. has as many critical points as the ho…

The Morse-Bott inequalities relate the topology of a closed manifold to the topology of the critical point set of a Morse-Bott function defined on it. The Morse-Bott inequalities are sometimes stated under incorrect orientation assumptions. We show that these assumptions are insufficient with an explicit counterexample…

We complete the theoretical framework required for the construction of a Morse homology theory for certain types of forced mean curvature flows. The main result of this paper describes the asymptotic behaviour of these flows as the forcing term tends to infinity in a certain manner. This result allows the Morse homolog…

We develop functoriality for Morse theory, namely, to a pair of Morse-Smale systems and a generic smooth map between the underlying manifolds we associate a chain map between the corresponding Morse complexes, which descends to the correct map on homology. This association does not in general respect composition. We gi…

The systole function has a universal index gap on moduli spaces.