Some Properties of Finitely Presented Groups with Topological Viewpoints

一种用于定义在合并树上的函数的有限稳定编辑距离

A Finitely Stable Edit Distance for Functions Defined on Merge Trees

本文提出一种比较不同合并树上函数的稳定编辑距离，可通过动态二进制线性规划计算，并在模拟数据上验证其适用于持久性图失效的TDA场景。

Lefschetz Complexes as Finite Topological Spaces

We consider a fixed basis of a finitely generated free chain complex as a finite topological space and we present a sufficient condition for the singular homology of this space to be isomorphic with the homology of the chain complex.

同调群拓扑的比较：从子群拓扑视角出发

Comparison of Topologies on Homotopy Groups with Subgroup Topology Viewpoint

本文从子群拓扑视角出发，比较不同拓扑结构在同调群上的应用，以揭示拓扑化同调群的相关结果。

Topological pullback, covering spaces, and a triad of Quillen

We study pullback from a topological viewpoint with emphasis on pullback of covering maps. We generalize a triad of Quillen on properties of the pullback functor.

On Topologized Fundamental Groups with Small Loop Transfer Viewpoints

In this paper, by introducing some kind of small loop transfer spaces at a point, we study the behavior of topologized fundamental groups with the compact-open topology and the whisker topology, $π_{1}^{qtop}(X,x_{0})$ and $π_{1}^{wh}(X,x_{0})$, respectively. In particular, we give necessary or sufficient conditions for coincidence and being topological group of these two topologized fundamental groups. Finally, we give some examples to show that the reverse of some of these implications do not hold, in general.

Topological complexity, fibrations and symmetry

We show how locally smooth actions of compact Lie groups on a manifold $X$ can be used to obtain new upper bounds for the topological complexity $\TC(X)$, in the sense of Farber. We also obtain new bounds for the topological complexity of finitely generated torsion-free nilpotent groups.

Persistent and Zigzag Homology: A Matrix Factorization Viewpoint

Over the past two decades, topological data analysis has emerged as a field of applied mathematics with new applications and algorithmic developments appearing rapidly. Two fundamental computations in this field are persistent homology and zigzag homology. In this paper, we show how these computations in the most general case reduce to finding a canonical form of a matrix associated with a type A quiver representation, which in turn can be computed using factorizations of associated matrices. We show how to use arbitrary induced maps on homology for computation, providing a framework that goes…

Nonabelian cohomology jump loci from an analytic viewpoint

For a topological space, we investigate its cohomology support loci, sitting inside varieties of (nonabelian) representations of the fundamental group. To do this, for a CDG (commutative differential graded) algebra, we define its cohomology jump loci, sitting inside varieties of (algebraic) flat connections. We prove that the analytic germs at the origin 1 of representation varieties are determined by the Sullivan 1-minimal model of the space. Under mild finiteness assumptions, we show that, up to a degree $q$, the two types of jump loci have the same analytic germs at the origins, when the s…

Fibrations between finite topological spaces

We study Hurewicz fibrations between finite T$_0$--spaces from a combinatorial viewpoint and give strong conditions that a continuous map between finite T$_0$--spaces must satisfy in order to be a Hurewicz fibration. We also show that there exists a strong relationship between Hurewicz fibrations between finite T$_0$--spaces and Grothendieck bifibrations. Finally we give several interesting examples that illustrate this theory and show that many of the assumptions of our results are necessary.

Index of a finitistic space and a generalization of the topological central point theorem

In this paper we prove that if $G$ is a p-torus (resp. torus) group acting without fixed points on a finitistic space X (resp. with finitely many orbit types), then the G-index $i_G(X) < 1$. Using this G-index we obtain a generalization of the Central Point Theorem and also of the Tverberg Theorem for any d-dimensional Hausdorff space.

Towards a dichotomy for the Reidemeister zeta function

We prove a dichotomy between rationality and a natural boundary for the analytic behavior of the Reidemeister zeta function for automorphisms of non-finitely generated torsion abelian groups and for endomorphisms of groups $\mathbb Z_p^d,$ where $\mathbb Z_p$ the group of p-adic integers. As a consequence, we obtain a dichotomy for the Reidemeister zeta function of a continuous map of a topological space with fundamental group that is non-finitely generated torsion abelian group. We also prove the rationality of the coincidence Reidemeister zeta function for tame endomorphisms pairs of finitel…

从算术拓扑角度看降阶理论与等变范畴

An Arithmetic Topology viewpoint on Descent theory and Equivariant Categories

本文通过算术拓扑视角，统一了降阶理论与等变范畴，建立了算术与拓扑不变量的类比，并展示了其在范畴层面的应用。

The Medusa of Spatial Sorting: Topological Construction

We consider the simultaneous movement of finitely many colored points in space, calling it a spatial sorting process. The name suggests a purpose that drives the collection to a configuration of increased or decreased order. Mapping such a process to a subset of space-time, we use persistent homology measurements of the time function to characterize the process topologically.

Topological quantum field theories and homotopy cobordisms

We construct a category $\mathrm{HomCob}$ whose objects are {\it homotopically 1-finitely generated} topological spaces, and whose morphisms are {\it cofibrant cospans}. Given a manifold submanifold pair $(M,A)$, we prove that there exists functors into $\mathrm{HomCob}$ from the full subgroupoid of the mapping class groupoid $\mathrm{MCG}_{M}^{A}$, and from the full subgroupoid of the motion groupoid $\mathrm{Mot}_{M}^{A}$, whose objects are homotopically 1-finitely generated. We also construct a family of functors $\mathsf{Z}_G\colon \mathrm{HomCob}\to \mathbf{Vect}$, one for each finite gro…