Stratified Spaces Research Articles

Stability of L2-Invariants on Stratified Spaces

Abstract Let $\overline{M}$ be a compact smoothly stratified pseudo-manifold endowed with a wedge metric $g$. Let $\overline{M}_{\Gamma }$ be a Galois $\Gamma $-covering. Under additional assumptions on $\overline{M}$, satisfied for example by Witt pseudo-manifolds, we show that the $L^{2}$-Betti numbers and the Novikov–Shubin invariants are well defined. We then establish their invariance under a smoothly stratified codimension-preserving homotopy equivalence, thus extending results of Dodziuk, Gromov, and Shubin to these pseudo-manifolds.

On the topologies of the exponential

Factorization algebras have been defined using three different topologies on the Ran space. We study these three different topologies on the exponential, which is the union of the Ran space and the empty configuration, and show that an exponential property is satisfied in each case. As a consequence, we describe the weak homotopy type of the exponential \mathrm{Exp}(X) for each topology, in the case where X is not (necessarily) connected.We also study these exponentials as stratified spaces and show that the metric exponential is conically stratified for a general class of spaces. As a corollary, we obtain that locally constant factorization algebras defined by Beilinson–Drinfeld are equivalent to locally constant factorization algebras defined by Lurie.

Characterizing some topological spaces with set-valued maps to a space with an original point

In this paper, the notion of an original point for a topological space is introduced. Then characterizations of some spaces such as stratifiable spaces and MCP-spaces in terms of set-valued maps with values in F(Y), where Y has an original point, are presented.

Gromov–Hausdorff convergence of metric pairs and metric tuples

We study the Gromov–Hausdorff convergence of metric pairs and metric tuples and prove the equivalence of different natural definitions of this concept. We also prove embedding, completeness and compactness theorems in this setting. Finally, we get a relative version of Fukaya's theorem about quotient spaces under Gromov–Hausdorff equivariant convergence and a version of Grove–Petersen–Wu's finiteness theorem for stratified spaces.

On the homotopy theory of stratified spaces

On the homotopy theory of stratified spaces

𝐾-theory of multiparameter persistence modules: Additivity

Persistence modules stratify their underlying parameter space, a quality that makes persistence modules amenable to study via invariants of stratified spaces. In this article, we extend a result previously known only for one-parameter persistence modules to grid multiparameter persistence modules. Namely, we show the K K -theory of grid multiparameter persistence modules is additive over strata. This is true for both standard monotone multi-parameter persistence as well as multiparameter notions of zig-zag persistence. We compare our calculations for the specific group K 0 K_0 with the recent work of Botnan, Oppermann, and Oudot, highlighting and explaining the differences between our results through an explicit projection map between computed groups.

Metric, stratifiable and uniform spaces of G-permutation degree

Metric, stratifiable and uniform spaces of <i>G</i>-permutation degree

Brittle bonds: queer relationalities in literary and cinematic forms in India

Abstract From factories to hostels, homes to neighbourhoods, how do embodied practices in stratified spaces shape the contemporary terrains of queerness in India? In this article, I analyse two recent cultural texts from India, Neeraj Ghaywan’s Geeli Pucchi (Sloppy Kisses, 2021) and Hansda Sowvender Shekhar’s (2018) novel My Father’s Garden, Speaking Tiger Books, Delhi, to engage with the fraught dynamics of relationality and queerness. My analysis contests any easy celebration of community – by foregrounding the movements of dispossessed protagonists who do not fit into the paradigm of the privileged subject of queer politics. Yet these texts capture the fleeting possibilities of reaching towards one another in spaces carved through the operations of exclusion and discrimination. I bring together two texts in which the direct references to sexual identities formed via governmental or non-governmental networks are largely absent; rather, they ask fundamental questions about the in-between terrain of the relational. The dynamics of distance and connection is opened up in complex ways in these cinematic and literary texts as they create imaginative idioms to explore the brittleness of queer bonds formed through the hierarchized operations of class, caste and ethnicity. Through an analysis of the formal aspects of these texts, and the practices of spectatorship and readership they facilitate, I seek to underline how formations of sexuality in India can be unpacked only through a close engagement with the critical discourse on gender, caste and ethnicity.

Homotopy truncations of homotopically stratified spaces

Intersection homology of Goresky and MacPherson can be defined from the Deligne sheaf, obtained from truncations of complexes of sheaves. As intersection homology is not the homology of a particular space, the search for a family of spaces whose homologies have properties analogous to intersection homology has developed. For some stratified spaces, M. Banagl has introduced such a family by using a topological truncation: the original link is replaced by a truncation of its homological Moore resolution. In this work, we study the dual approach in the Eckmann-Hilton sense: we consider the stratified space obtained by replacing the original link by a Postnikov approximation. The main result is that our construction restores the space constructed by Gajer to establish an intersection Dold-Thom theorem. We are conducting this study within the general framework of Quinn’s homotopically stratified spaces.

Functorial Properties of the Space for Stratifiable Spaces X

In the category of stratifiable spaces and continuous mappings into itself, we bring one construction belonging to Coty called test space, which defines a covariant functor in this category. This construction defines a functor , which allows each stratifiable space to be immersed in a closed manner into some other space which is a stratifiable space with "Good" functorial, geometric and topological properties. It is shown that the functor is a normal, open and monadic functor in this category of stratifiable spaces and continuous mappings into itself. And also it is studied the dimensional properties of the space for the stratifiable space , defined for each subfunctor of the functor for which the dimension satisfies the inequality: .

Posets for which Verdier duality holds

We discuss two known sheaf-cosheaf duality theorems: Curry’s for the face posets of finite regular CW complexes and Lurie’s for compact Hausdorff spaces, i.e., covariant Verdier duality. We provide a uniform formulation for them and prove their generalizations. Our version of the former works over the sphere spectrum and for more general finite posets, which we characterize in terms of the Gorenstein* condition. Our version of the latter says that the stabilization of a proper separated infty -topos is rigid in the sense of Gaitsgory. As an application, for stratified topological spaces, we clarify the relation between these two duality equivalences.

L^p-Cohomology, Heat Semigroup and Stratified Spaces

Let (M, g) be an incomplete Riemannian manifold of finite volume and let 2le p<infty . In the first part of this paper we prove that under certain assumptions the inclusion of the space of L^p-differential forms into that of L^2-differential forms gives rise to an injective/surjective map between the corresponding L^p and L^2 cohomology groups. Then in the second part we provide various applications of these results to the curvature and the intersection cohomology of compact Thom–Mather stratified pseudomanifolds and complex projective varieties with only isolated singularities.

A general categorical reflection for various completions of [formula omitted]-closure spaces and [formula omitted]-ordered sets

In this paper, we consider a full subcategory S of the category CLQ of stratified Q-closure spaces, satisfying some conditions. We construct a reflection from the category CLQ onto S, called the S-completion, which encompasses a variety of completions for Q-closure spaces, Q-cotopological spaces, and Q-ordered sets. The notion of a fuzzy subset system Z on a category C of structured sets is introduced. By considering fuzzy subset systems Z on the category CLQ, we introduce the concept of a Z-convergence Q-closure space, as a generalization of sober/bounded sober/monotone convergence Q-cotopological spaces. We show that the category S is exactly a category of Z-convergence spaces for some fuzzy subset system Z. And, in the case that Z consists of irreducible fuzzy sets, the S-completion can be restricted to the setting of Q-cotopological spaces. The sobrification, bounded sobrification, D-completion of Q-cotopological spaces are special cases of the S-completion. After establishing the connections between fuzzy subset systems on CLQ and those on the category ORDQ of Q-ordered sets, we provide a general approach to generating various completions of Q-ordered sets from the S-completion of corresponding Q-closure spaces. Completions of Q-ordered sets with respect to different types of fuzzy directedness can be obtained directly in this useful framework.

Whitney stratifications are conically smooth

The notion of conically smooth structure on a stratified space was introduced by Ayala, Francis and Tanaka. This is a very well behaved analogue of a differential structure in the context of stratified topological spaces, satisfying good properties such as the existence of resolutions of singularities and handlebody decompositions. In this paper we prove Ayala, Francis and Tanaka’s conjecture that any Whitney stratified space admits a canonical conically smooth structure. We thus establish a connection between the theory of conically smooth spaces and the classical examples of stratified spaces from differential topology.

Polyhedral compactifications, I

Abstract In this work we describe horofunction compactifications of metric spaces and finite-dimensional real vector spaces through asymmetric metrics and asymmetric polyhedral norms by means of nonstandard methods, that is, by ultrapowers of the spaces at hand. The polyhedral compactifications of the vector spaces carry the structure of stratified spaces with the strata indexed by dual faces of the polyhedral unit ball. Explicit neighborhood bases and descriptions of the horofunctions are provided.

H-Principle for Stratified Spaces

Abstract We extend Gromov and Eliashberg–Mishachev’s $h-$principle on manifolds to stratified spaces. This is done in both the sheaf-theoretic framework of Gromov and the smooth jets framework of Eliashberg–Mishachev. The generalization involves developing 1. the notion of stratified continuous sheaves to extend Gromov’s theory, 2. the notion of smooth stratified bundles to extend Eliashberg–Mishachev’s theory. A new feature is the role played by homotopy fiber sheaves. We show, in particular, that stratumwise flexibility of stratified continuous sheaves along with flexibility of homotopy fiber sheaves furnishes the parametric $h$-principle. We extend the Eliashberg–Mishachev holonomic approximation theorem to stratified spaces. We also prove a stratified analog of the Smale–Hirsch immersion theorem.

Long time existence of Yamabe flow on singular spaces with positive Yamabe constant

In this work we establish long-time existence of the normalized Yamabe flow with positive Yamabe constant on a class of manifolds that includes spaces with incomplete cone-edge singularities. We formulate our results axiomatically, so that our results extend to general stratified spaces as well, provided certain parabolic Schauder estimates hold. The central analytic tool is a parabolic Moser iteration, which yields uniform upper and lower bounds on both the solution and the scalar curvature.

Differentiable stratified groupoids and a de Rham theorem for inertia spaces

We introduce the notions of differentiable groupoids and differentiable stratified groupoids, generalizations of Lie groupoids in which the spaces of objects and arrows have the structures of differentiable spaces, respectively differentiable stratified spaces, compatible with the groupoid structure. After studying basic properties of these groupoids including Morita equivalence, we prove a de Rham theorem for proper locally contractible differentiable stratified groupoids. We then focus on the study of the inertia groupoid associated to a proper Lie groupoid. We show that the loop and the inertia space of a proper Lie groupoid can be endowed with a natural Whitney (b)-regular stratification, which we call the orbit Cartan type stratification. Endowed with this stratification, the inertia groupoid of a proper Lie groupoid becomes a locally contractible differentiable stratified groupoid.

Orbifolds as stratified diffeologies

We discuss general properties of stratified spaces in diffeology. This leads to a formal framework for the theory of stratifications. In particular, we consider the Klein stratification of diffeological orbifolds, defined by the action of local diffeomorphisms. We show that it is a standard stratification in the sense that the partition of the space into orbits of local diffeomorphisms is locally finite (for orbifolds with locally finite atlases), it satisfies the frontier condition and the orbits are locally closed manifolds.

Function characterizations of some topological spaces

We present some new characterizations of some spaces such as stratifiable spaces, MCP-spaces in terms of real-valued functions.