Related papers: Discrete Morse theory on $\Omega S^2$
There are two rather distinct approaches to Morse theory nowadays: smooth and discrete. We propose to study a real valued function by assembling all associated sections in a topological category. From this point of view, Reeb functions on…
Following \cite{citeSavelyevVirtualMorsetheoryon$Omega$Ham$(Momega)$.}, we develop here a connection between Morse theory for the (positive) Hofer length functional $L: \Omega \text {Ham}(M, \omega) \to \mathbb{R}$, with Gromov-Witten/Floer…
In this paper we discuss the topology of the symplectomorphism group of a product of two 2-dimensional spheres when the ratio of their areas lies in the interval (1,2]. More precisely we compute the homotopy type of this symplectomorphism…
We provide a recursive construction of an acyclic matching (also known as a gradient vector field, an equivalent notion to a discrete Morse function) on the independence complex of a graph with a simplicial vertex using given acyclic…
We study a spectral sequence that computes the (mod 2) S^1-equivariant homology of the free loop space LM of a manifold M (the "string homology" of M). Using it and knowledge of the string topology operations on the homology of LM, we…
In this paper, we determine the homotopy type of the Morse complex of certain collections of simplicial complexes by studying dominating vertices or strong collapses. We show that if $K$ contains two leaves that share a common vertex, then…
Let $K$ be an arbitrary semi-cubical set that can be embedded in a standard cube. Using Discrete Morse Theory, we construct a CW-complex that is homotopy equivalent to the space $\vec{P}(K)_v^w$ of directed paths between two given vertices…
In this paper, we develop the notion of a Morse sequence, which provides an alternative approach to discrete Morse theory, and which is both simple and effective. A Morse sequence on a finite simplicial complex is a sequence composed solely…
The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of mass represents a rather restricted class M_C of Morse-Smale functions on S^2.…
The aim of this paper is to give a (discrete) Morse theoretic proof of the fact that the $k$-th skeleton of the flag complex $\mathcal{F}$, associated to the lattice of subspaces of a finite dimensional vector space, is homotopy equivalent…
A $2$-matching complex is a simplicial complex which captures the relationship between $2$-matchings of a graph. In this paper, we will use discrete Morse Theory and the Matching Tree Algorithm to prove homotopical results. We will consider…
In this paper, we construct a pointed CW complex called the magnitude homotopy type for a given metric space $X$ and a real parameter $\ell \geq 0$. This space is roughly consisting of all paths of length $\ell$ and has the reduced homology…
We develop a new method to compute the homology groups of finite topological spaces (or equivalently of finite partially ordered sets) by means of spectral sequences giving a complete and simple description of the corresponding…
We use the framework of Morse theory with differential graded coefficients to study certain operations on the total space of a fibration. More particularly, we focus in this paper on a chain-level description of the Chas-Sullivan product on…
Let $f$ be a Morse function on a smooth compact manifold $M$ with boundary. The path component $\mathrm{PH}_f^{-1}(D)$ containing $f$ of the space of Morse functions giving rise to the same Persistent Homology $D=\mathrm{PH}(f))$ is shown…
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…
Persistent Homology (PH) allows tracking homology features like loops, holes and their higher-dimensional analogs, along with a single-parameter family of nested spaces. Currently, computing descriptors for complex data characterized by…
We prove that group homology of the diffeomorphism group of $\#^g S^n \times S^n$ as a discrete group is independent of $g$ in a range, provided that $n>2$. This answers the high dimensional version of a question posed by Morita about…
Recent work on homotopy type theory exploits an exciting new correspondence between Martin-Lof's dependent type theory and the mathematical disciplines of category theory and homotopy theory. The category theory and homotopy theory suggest…
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…