In a previous work we proved the uniqueness and functoriality of primary unfoldings on simple Thom-Mather spaces, which is a functor to the category of smooth manifolds. In this article we extend these results for any stratified Thom-Mather pseudomanifold with arbitary finite length, through a new kind of intermediate …
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48 results for “Thom-Mather stratified space”
This article is devoted to the study of smooth desingularization, which are customary employed in the definition of De Rham Intersection Cohomology with differential forms. In this paper we work with the category of Thom-Mather simple spaces. We construct a functor which sends each Thom-Mather simple space into a smoot…
The main result of this paper is a sufficient condition in order to have a compact Thom-Mather stratified pseudomanifold endowed with a -iterated edge metric on its regular part -parabolic. Moreover, besides stratified pseudomanifolds, the -parabolicity of other classes of singular spaces, such as compac…
Extends cohomology to incomplete Riemannian manifolds.
problem Cohomology of harmonic forms on incomplete Riemannian manifolds.
method Abstract setting of Hilbert complexes.
result Geometric applications to Thom-Mather spaces.
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 …
Constructs a Dirac-Schrödinger operator on Thom-Mather stratified spaces.
problem Computing an index for orbit spaces of semi-free -actions.
method Pushing down an -invariant first order transversally elliptic operator to the quotient space.
result The constructed operator's index coincides with Lott's signature.
The paper studies cohomology on incomplete manifolds and stratified spaces.
problem Analyzing cohomology groups on incomplete Riemannian manifolds and stratified spaces.
method Proves injective/surjective maps between and cohomology groups under certain conditions.
result Injective/surjective maps between and cohomology groups are established.
We develop a generalization to non-Witt spaces of the intersection homology theory of Goresky-MacPherson. The second author has described the self-dual sheaves compatible with intersection homology, and the other authors have described a generalization of Cheeger's L2 de Rham cohomology. In this paper we extend both of…
New method defines Gysin maps for stratified spaces, preserving signatures.
problem Defining and preserving signatures in stratified spaces.
method Smooth atlas stratified spaces, fiber bundles, Gysin maps, KK-theory.
result Gysin maps preserve analytic signature classes of Witt spaces.
Positive Scalar Curvature on Spin Pseudomanifolds: the Fundamental Group and Secondary Invariantsmath.DG
Study on positive scalar curvature on complex spaces with specific conditions.
problem Existence and sufficiency of positive scalar curvature metrics on stratified spaces.
method Index-theoretic conditions and cobordism analysis.
result Sufficient conditions for positive scalar curvature on certain stratified spaces.
The paper explores conditions for positive scalar curvature on specific stratified spaces.
problem Conditions for the existence of a wedge metric with positive scalar curvature.
method Relates the problem to index theory on stratified spaces and uses obstruction and sufficient conditions.
result Sufficient conditions for the existence of a wedge metric with positive scalar curvature.
Given a compact stratified pseudomanifold with a Thom-Mather stratification and a class of riemannian metrics over its regular part, we study the relationships between the de Rham and Hodge cohomology and the intersection cohomology of associated to some perversities. More precisely, to a kind of metric whi…
Study geometric equivalence of smooth map germs.
problem Equivalence relations among smooth map germs with respect to G-structures.
method Generalization of right-left equivalence (A-equivalence) to include geometric structures.
result Interesting applications of these equivalence relations.
The paper (in French) exemplifies graphically a solution of the heat equation which is a 1-dimensional unfolding of an elliptic umbilic catastrophe. The example is due to James Damon and adapts Thom-Mather's singularity theory to multiscale models of scale-space analysis in image processing.
Develops a Thom-Mather theory for corank 1 frontals.
problem Characterizing and constructing stable unfoldings of corank 1 frontals.
method Thom-Mather theory, frontal stability, versality of unfoldings, reduction method.
result Classification of stable frontal multigerms and corank 1 stable frontals.
Symplectic embedding extended to stratified spaces.
problem Symplectic embedding theorem for stratified spaces.
method Defined symplectic structure on stratified spaces, demonstrated embedding in complex projective space.
result Symplectic embedding theorem extended to stratified spaces.
The paper explores density of stable mappings and their properties.
problem Density of stable mappings in different dimensions.
method Infinitesimal and algebraic methods to prove density of proper stable and topologically stable mappings.
result Density of topologically stable mappings holds for any pair (n,p), and for proper stable mappings if (n,p) is in nice dimensions.
Paper introduces stratified vector bundles and their properties.
problem Understanding singular spaces and their vector bundles.
method Characterization via monoid actions and examples from various fields.
result Functorial properties extended to the stratified case.
Paper proves Whitney stratified spaces can be given a conically smooth structure.
problem Proving Whitney stratified spaces can be given a conically smooth structure.
method Introduced conically smooth structure by Ayala, Francis, and Tanaka. Proved conjecture that any Whitney stratified space admits a canonical conically smooth structure.
result Established a connection between Whitney stratified spaces and conically smooth spaces.
Extends h-principle to stratified spaces using sheaf and jet theories.
problem Applying h-principle to stratified spaces.
method Developed new sheaf and bundle theories for stratified spaces, and proved the h-principle.
result Stratified continuous sheaves and homotopy fiber sheaves lead to the parametric h-principle.
The paper introduces knot invariants using stratified homotopy groups.
problem Defining knot invariants in stratified spaces.
method Introducing stratified homotopy groups and proving their properties.
result Stratified Whitehead's theorem holds for these groups.
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.
Sphere theorems extended to RCD spaces and improved for Einstein stratified spaces.
problem Generalizing sphere theorems to new types of spaces.
method Proved sphere theorems for RCD(n-1, n) spaces and Einstein stratified spaces.
result Extended sphere theorems to RCD spaces and improved results for Einstein stratified spaces.
In this paper we introduce the notion of a smooth structure on a stratified space, the notion of a Poisson smooth structure and the notion of a weakly symplectic smooth structure on a stratified symplectic space, refining the concept of a stratified symplectic Poisson algebra introduced by Sjamaar and Lerman. We show t…
Jet spaces on Carnot groupsmath.DG
Jet spaces on Carnot groups have a canonical Lie group structure.
problem Understanding jet spaces on Carnot groups.
method Constructing jet spaces over stratified Lie groups and showing they are stratified Lie groups.
result Every stratified Lie group of step can be embedded in a jet space over a stratified Lie group of step .
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…
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…
Studied invariants on stratified spaces, proving their stability.
problem Stability of invariants on stratified spaces.
method Defined and analyzed -Betti numbers and Novikov-Shubin invariants for compact smoothly stratified pseudo-manifolds with a wedge metric, extending results to these pseudo-manifolds.
result Invariance of -Betti numbers and Novikov-Shubin invariants under smoothly stratified, strongly stratum preserving homotopy equivalence.
Proves Verdier duality for sheaves on stratified spaces.
problem Verdier duality for constructible sheaves on stratified spaces.
method Uses conically smooth stratified spaces and Lurie's Verdier duality.
result Shows equivalence between constructible sheaves and cosheaves.
We show that conically smooth stratified spaces embed fully faithfully into -categories. This articulates a stratified generalization of the homotopy hypothesis proposed by Grothendieck. As such, each -category defines a stack on conically smooth stratified spaces, and we identify the descent conditions…
We introduce the notions of a differentiable groupoid and a differentiable stratified groupoid, 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 b…
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…
Study of hyperkahler spaces from Lie algebras.
problem Understanding hyperkahler quotients of cotangent bundles.
method Analyzing stratified spaces of cotangent bundles of complex semisimple Lie groups.
result Explicit description of partial order on strata using Lie theoretic data.
The paper extends higher signature invariants to stratified spaces and establishes a new surgery exact sequence.
problem Extending higher signature invariants to stratified spaces.
method Revisiting and extending the surgery exact sequence for stratified spaces.
result Established a natural transformation between the surgery exact sequence and a long exact sequence of K-theory groups for stratified spaces.
Skeleta and other pure subsets of manifold stratified spaces are shown to have neighborhoods which are teardrops of stratified approximate fibrations (under dimension and compactness assumptions). In general, the stratified approximate fibrations cannot be replaced by bundles, and the teardrops cannot be replaced by ma…
Classification of locally standard -pseudomanifolds over topological stratified pseudomanifoldsmath.GT
Paper classifies pseudomanifolds over stratified spaces.
problem Classifying pseudomanifolds over stratified spaces.
method Introducing locally standard -pseudomanifolds and using characteristic data.
result Locally standard -pseudomanifolds over topological stratified pseudomanifolds are classified by their characteristic data.
The paper studies control data and equivariant properties of stratified spaces.
problem Equivariant properties of stratified spaces with group actions.
method Equivariant submersion theorem and control data construction.
result Inclusion of -stratified subspaces is a -equivariant cofibration.
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 …
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.
Control data constructed for smooth weak deformation retraction of stratified spaces.
problem Construct control data for smooth weak deformation retraction of stratified spaces.
method Show smooth local triviality with conical fibers, construct control data, use fiber-wise scalar multiplications.
result Obtain neighbourhood smooth weak deformation retraction of stratified spaces.
Gauss-Green theorem proven for vector fields in stratified groups.
problem Establishing the Gauss-Green theorem for vector fields in noncommutative stratified Lie groups.
method Developed a new family of function spaces for divergence-measure fields and proved the Gauss-Green theorem.
result Gauss-Green theorem achieved for vector fields of low regularity on sets of finite perimeter in stratified groups.
New definition of regular points for PL functions on manifolds.
problem Defining regular points for PL functions on combinatorial manifolds.
method Definition based on link of the point, stratification of Jacobi set, Stein factorization of Reeb space.
result Our definition of regularity is distinct from existing definitions.
We show that intersection homology extends Poincare duality to manifold homotopically stratified spaces (satisfying mild restrictions). This includes showing that, on such spaces, the sheaf of singular intersection chains is quasi-isomorphic to the Deligne sheaf.
Stratified fibre bundlesmath.GT
A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theor…
Filling invariants are measurements of a metric space describing the behaviour of isoperimetric inequalities. In this article we examine filling functions and higher divergence functions. We prove for a class of stratified nilpotent Lie groups that in the low dimensions the filling functions grow as fast as the ones of…
Consider a stratified space with a positive Ricci lower bound on the regular set and no cone angle larger than 2. For such stratified space we know that the first non-zero eigenvalue of the Laplacian is larger than or equal to the dimension. We prove here an Obata rigidity result when the equality is attained: the l…
Formula for index of Dirac-type operators on stratified spaces.
problem Calculating the index of Dirac-type operators on complex geometric structures.
method Defined a closed domain, proved self-adjoint and Fredholm properties, established index formula.
result Proved a formula for the Chern character of the index of Dirac-type operators.
In this article we prove that stratified spaces and other geometric subfamilies satisfy categorical Fraïssé properties, a matter that might be of interest for both geometers and logicians. As a motivation we show a new example of a stratified pseudomanifold that satisfies the finite oscillation property with respect to…