arXiv research

Extends h-principle to stratified spaces using sheaf and jet theories.

By considering homotopies that preserve the stratification, one obtains a natural notion of homotopy for stratified spaces. In this short note, we introduce invariants of stratified homotopy, the stratified homotopy groups. We show that they satisify a stratified version of Whitehead's theorem. As an example, we introd…

New homotopy theory reveals the structure of stable curves.

The paper explores conditions for compactness and finiteness in stratified homotopy theory.

We show that conically smooth stratified spaces embed fully faithfully into $\infty$-categories. This articulates a stratified generalization of the homotopy hypothesis proposed by Grothendieck. As such, each $\infty$-category defines a stack on conically smooth stratified spaces, and we identify the descent conditions…

We propose a new notion of `n-category with duals', which we call a Whitney n-category. There are two motivations. The first is that Baez and Dolan's Tangle Hypothesis is (almost) tautological when interpreted as a statement about Whitney categories. The second is that we can functorially construct `fundamental Whitney…

We prove the Novikov conjecture on oriented Cheeger spaces whose fundamental group satisfies the strong Novikov conjecture. A Cheeger space is a stratified pseudomanifold admitting, through a choice of ideal boundary conditions, an L2-de Rham cohomology theory satisfying Poincare duality. We prove that this cohomology …

Link concordance equals homotopy for high-dimensional spheres.

Studied $L^2-$invariants on stratified spaces, proving their stability.

In work of Higson-Roe the fundamental role of the signature as a homotopy and bordism invariant for oriented manifolds is made manifest in how it and related secondary invariants define a natural transformation between the (Browder-Novikov-Sullivan-Wall) surgery exact sequence and a long exact sequence of C*-algebra K-…

This is a sequel to the paper "The signature package on Witt spaces, I. Index classes" by the same authors. In the first part we investigated, via a parametrix construction, the regularity properties of the signature operator on a stratified Witt pseudomanifold, proving, in particular, that one can define a K-homology …

We show that the set of harmonic maps from the 2-dimensional stratified spheres with uniformly bounded energies contains only finitely many homotopy classes. We apply this result to construct infinitely many harmonic map flows and mean curvature flows of 2-sphere in the connected sum of two closed 3-dimensional manifol…

Simplified 3D Dijkgraaf-Witten theory with defects explained geometrically.

Paper introduces stratified vector bundles and their properties.

We study bordism groups and bordism homology theories based on pseudomanifolds and stratified pseudomanifolds. The main seam of the paper demonstrates that when we uses classes of spaces determined by local link properties, the stratified and unstratified bordism theories are identical; this includes the known examples…

In this paper, we develop Leray-Serre-type spectral sequences to compute the intersection homology of the regular neighborhood and deleted regular neighborhood of the bottom stratum of a stratified PL-pseudomanifold. The E^2 terms of the spectral sequences are given by the homology of the bottom stratum with a local co…

We study torsion properties of the twisted Alexander modules of the affine complement $M$ of a complex essential hyperplane arrangement, as well as those of punctured stratified tubular neighborhoods of complex essential hyperplane arrangements. We investigate divisibility properties between the twisted Alexander polyn…

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…

Character varieties of knot groups into SU(3) are stratified and analyzed.

In a previous article the author extended the Witten deformation to singular spaces with cone-like singularities and to a class of Morse functions called admissible Morse functions. The method applies in particular to complex cones and stratified Morse functions in the sense of the theory developed by Goresky and MacPh…

We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the cla…

Paper proves Whitney stratified spaces can be given a conically smooth structure.

We describe an algorithm that associates to each positive real number $r$ and each finite collection $C_r$ of planar pixels of size $r$ a planar piecewise linear set $S_r$ with the following additional property: if $C_r$ is the collection of pixels of size $r$ that touch a given compact semialgebraic set $S$, then the …

Witt spaces are pseudomanifolds for which the middle-perversity intersection homology with rational coefficients is self-dual. We give a new construction of the symmetric signature for Witt spaces which is similar in spirit to the construction given by Miscenko for manifolds. Our construction has all of the expected pr…

This article is a survey of recent work of the author, together with Markus Banagl, Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza, on the Hodge theory of stratified spaces. We discuss how to resolve a Thom-Mather stratified space to a manifold with corners with an iterated fibration structure and the generalization of …

This work forms a foundational study of factorization homology, or topological chiral homology, at the generality of stratified spaces with tangential structures. Examples of such factorization homology theories include intersection homology, compactly supported stratified mapping spaces, and Hochschild homology with c…

Authors classify 3D locally standard T-pseudomanifolds under weaker conditions.

This paper surveys the significant progress over the past couple of decades in the theory of stratified spaces through the application of controlled methods as well as through the application of intersection homology.

Jet spaces on Carnot groups have a canonical Lie group structure.

The paper explores holonomy, zeta functions, and cohomology in foliated manifolds with stratified boundaries.

We lay the foundations for a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups. Such vector fields form a new family of function spaces, which generalize in a sense the $BV$ fields. They provide the most general setting to establish Gauss-Green formulas for vector fields of low regul…

The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, González, and Rudyak [1009.1851] with the aim of constructing a cellular model of the configuration space of a sphere. In particular, it was shown that the classifying space (order complex) of the face poset of a totally …

Study minimax estimation of stratified structure from i.i.d. samples.

In this paper, we explore the theme of orbifold stratified spaces and establish a general criterion for them to be smooth orbifolds. This criterion utilizes the notion of linear stratification on the gluing bundles for the orbifold stratified spaces. We introduce a concept of good gluing structure to ensure a smooth st…

Develops analytic methods for Lefschetz and Morse theories on stratified pseudomanifolds.

The monodromy action in the homology of level sets of Morse functions on stratified singular analytic varieties is studied. The local variation operators in both the standard and the intersection homology groups defined by the loops around the critical values of such functions are reduced to similar operators in the ho…

In this paper we prove a variety of results about the signature operator on Witt spaces. First, we give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold X which satisfies the Witt condition. This construction, which is inductive over the `depth' of the singularity…

A stratified Lie system is a nonautonomous system of first-order ordinary differential equations on a manifold $M$ described by a $t$-dependent vector field $X=\sum_{α=1}^rg_αX_α$, where $X_1,\ldots,X_r$ are vector fields on $M$ spanning an $r$-dimensional Lie algebra that are tangent to the strata of a stratification …

Study on scalar curvature in wedge spaces with existence and obstruction results.

New method defines Gysin maps for stratified spaces, preserving signatures.

The purpose of this paper is to study finite dimensional equivariant moduli problems from the viewpoint of stratification theory. We show that there exists a stratified obstruction system for a finite dimensional equivariant moduli problem. In addition, we define a coindex for a G-vector bundle which is determined by t…

We study the spaces of polynomials stratified into the sets of polynomial with fixed number of roots inside certain semialgebraic region $Ω$, on its border, and at the complement to its closure. Presented approach is a generalisation, unification and development of several classical approaches to stability problems in …

Minimal surfaces with complex branching structures constructed using various methods.

Eigen-stratified models reduce model size and improve performance.

We derive spectral sequences for the intersection homology of stratified fibrations and approximate tubular neighborhoods in manifold stratified spaces. These neighborhoods include regular neighborhoods in PL stratified spaces.

Symplectic embedding extended to stratified spaces.

V. Turaev introduced the theory of topology of words and phrases in 2005. This is a combinatorialy extension of the theory of virtual knots and links. In this paper we generalize the notion of homotopy of words and phrases and we give geometric meanings of the generalized homotopy of words. Moreover using the generaliz…

Extends geometric quantization to singular spaces.