Related papers: Elementary Methods for Persistent Homotopy Groups
Nilpotency for discrete groups can be defined in terms of central extensions. In this paper, the analogous definition for spaces is stated in terms of principal fibrations having infinite loop spaces as fibers, yielding a new invariant we…
We study some properties of A^1-homotopy groups: geometric interpretations of connectivity, excision results, and a re-interpretation of quotients by free actions of connected solvable groups in terms of covering spaces in the sense of…
We survey several mathematical developments in the holonomy approach to gauge theory. A cornerstone of this approach is the introduction of group structures on spaces of based loops on a smooth manifold, relying on certain homotopy…
Persistent homology is a topological data analysis tool that has been widely generalized, extending its scope beyond the field of topology. Among its extensions, steady and ranging persistence were developed to study a wide variety of graph…
In non-Hermitian physics, high-order exceptional points(HOEPs) with eigenvalues and eigenvectors coalesce are known for their enhanced sensitivity to perturbations. Typically, they exhibit eigenvalue splitting that scales as…
Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and…
We develop an axiomatic framework for persistent homology in any degree. We prove the existence and uniqueness for both a persistent version of the Eilenberg-Steenrod axioms for classical homology and a reduced version of this set of…
Persistent homology is a relatively new tool often used for \emph{qualitative} analysis of intrinsic topological features in images and data originated from scientific and engineering applications. In this paper, we report novel…
Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features,…
The minimal Quillen model is a free Lie model for rational spaces proposed by Quillen. Meanwhile, persistence modules are theoretical abstractions of persistent homology. In this paper, we integrate the ideas of rational homotopy theory and…
We study the grokking phenomenon through the lens of topology. Using persistent homology on point clouds derived from the embedding matrices of a range of models trained on modular arithmetic with varying primes, we identify a clear and…
The topological fundamental group $\pi_{1}^{top}$ is a homotopy invariant finer than the usual fundamental group. It assigns to each space a quasitopological group and is discrete on spaces which admit universal covers. For an arbitrary…
In this paper we present a new approach to computing homology (with field coefficients) and persistent homology. We use concepts from discrete Morse theory, to provide an algorithm which can be expressed solely in terms of simple graph…
In this master thesis, we extend results from classical simple homotopy theory to the world of stratified homotopy theory. To obtain a well-established framework to work in, we prove a series of results on two model categories of simplicial…
Topological complexity is a homotopy invariant that measures the minimal number of continuous rules required for motion planning in a space. In this work, we introduce persistent analogs of topological complexity and its cohomological lower…
In the uniformly discrete case of virtual persistence diagram groups $K(X,A)$, we construct a translation-invariant heat semigroup. The kernels are supported on a countable subgroup $H$, and the restriction to $H$ has Fourier exponent…
We define a category of filtered topological spaces and explore some of its homotopy theoretic properties, including a filtered analogue of CW approximation. With this, we define and study a filtered (weighted) variant of the Euler…
The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of…
D. K. Biss (Topology and its Applications 124 (2002) 355-371) introduced the topological fundamental group and presented some interesting basic properties of the notion. In this article we intend to extend the above notion to homotopy…
Given a manifold $M$ and a proper sub-bundle $\Delta\subset TM$, we study homotopy properties of the horizontal base-point free loop space $\Lambda$, i.e. the space of absolutely continuous maps $\gamma:S^1\to M$ whose velocities are…