Questions tagged [stratifications]

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How to chart tubes around manifolds with boundary/corners?

Let $M \subset \mathbb{R}^d$ be a manifold with boundary/corners. For example, a piece of curve with endpoints or a $2d$ unit square in $\{ z = 0 \}$. I am interested in introducing local coordinates ...
tsnao's user avatar
  • 462
2 votes
0 answers
105 views

Extension of isotopies

In what follows $M$ will be a manifold (without boundary, for simplicity) and $C\subseteq M$ will denote a compact subset. In the paper Deformations of spaces of imbeddings Edwards and Kirby prove the ...
Tommaso Rossi's user avatar
5 votes
0 answers
46 views

If a subset $X$ of a $C^k$ manifold $M$ is semialgebraic in the charts of $M$, is it Whitney stratifiable?

Let $M$ be a $C^k$ manifold for some $k\geq 1$ and $X$ be a subset of $M$. Assume that there is an atlas of charts $(\phi_\alpha, U_\alpha)_\alpha$ of $M$ such that in the coordinates of each of these ...
Charles Arnal's user avatar
5 votes
0 answers
100 views

Torsion in the spectral sequence for a stratified complex variety

Let $X$ be a (possibly singular) complex projective algebraic variety, endowed with a stratification $\{X_{\Delta}\}_{\Delta\in I}$ by smooth algebraic varieties. Then there is a spectral sequence $$...
Emiliano Ambrosi's user avatar
1 vote
1 answer
113 views

Smooth extension of piecewise smooth function on a corner

Imprecise Question: Suppose I have a function defined on non-codimension-zero strata of a smooth manifold with a stratification, and I know the function is smooth when restricted to each of these ...
Whitney Junior's user avatar
5 votes
0 answers
120 views

Under what assumption on a proper map does the preimage of sufficiently small neighborhood is homotopy equivalent to the fiber?

Let $\pi\colon X\rightarrow Y$ be a proper map of topological spaces. Let's assume that both $X$ and $Y$ are paracompact, Hausdorff and locally weakly contractible. Then is it enough to conclude that ...
user42024's user avatar
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5 votes
0 answers
82 views

Stacks v.s. Stratifolds?

Both stacks and stratifolds can model spaces with singularities (say, singular quotient space for example). In my little experience, stacks are widely used in moduli problems and stratifolds are often ...
Ruizhi liu's user avatar
1 vote
0 answers
27 views

Connected components of Isotropy types as strata of Poisson leaves

Let $X$ be a smooth affine variety with an algebraic symplectic form $\omega$. Let $G$ be a finite subgroup of the group of symplectomorphisms of $X$. We can say that $X$ is trivially a normal variety ...
Flavius Aetius's user avatar
1 vote
0 answers
67 views

Smooth affine variety as a symplectic resolutions

Given a smooth affine variety $X$ over $\mathbb C$ with an algebraic symplectic form $\omega$ and a finite group $G$ acting on $X$ by symplectomorphisms, then Is it true that $X$ is trivially a ...
Flavius Aetius's user avatar
6 votes
1 answer
350 views

Exit path categories of regular CW complexes

Given a finite, regular CW complex $X$ (by regular, I mean that the gluing maps $D^n \to X$ from the closed unit ball to $X$ are homeomorphisms onto their image), denote by $S$ the finite partially ...
Markus Zetto's user avatar
3 votes
0 answers
136 views

Topology types in families of real or complex varieties

In René Thom, "Structural Stability and Morphogenesis" on p. 21ff there is the following statement: Let $$P_j(x_i,s_k) = 0$$ be a set of polynomial equations over the real or complex numbers,...
Jürgen Böhm's user avatar
2 votes
1 answer
220 views

Comparing the exit path category and the nerve of a stratified space

Let $P$ be a finite poset, and $X$ be a topological space stratified by $P$, in the sense that $X$ is equipped with a continuous map $X \to P$ in the Alexandroff topology (or equivalently with a ...
Phil Tosteson's user avatar
2 votes
1 answer
128 views

On the zero-dimensional strata of the Fulton-MacPherson conpactification

Let $\operatorname{Conf}_n(\mathbb{R})$ be the configuration space of $n$ marked points on the real line. What is the difference between $\operatorname{Conf}_n(\mathbb{R})$ and the locus of zero-...
Banana23's user avatar
3 votes
0 answers
128 views

Riemann-Hilbert-type correspondence for locally constant factorization algebras

This is related to a previous post, but a bit softer and should probably stand on its own. In Appendix A of "Higher Algebra", Lurie shows that for a reasonably good topological space, there ...
Markus Zetto's user avatar
4 votes
0 answers
233 views

Blow-up of a stratified space

Let $X$ be a smooth projective variety over $\mathbb{C}$, and $D_1, \ldots, D_n$ be a collection of simple normal crossing divisors. The divisors induce a stratification $\mathcal{T}_X$ of $X$. Let $...
calc's user avatar
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3 votes
0 answers
211 views

Factorization algebras as factorizable cosheaves on the (extended) Ran Space

A basic fact in the theory of factorization algebras is that, to state it in a rough way, the exit path category of the Ran space of a topological manifold $M$ is equivalent to the category consisting ...
Markus Zetto's user avatar
6 votes
1 answer
163 views

Seeking a Weyl tube formula for Whitney stratified spaces

Background: Let $X$ be a smooth, compact Riemannian submanifold of euclidean space $\mathbb{R}^n$. H Weyl's tube formula asserts that for sufficiently small $t > 0$, the volume $V(X;t)$ of the ...
Vidit Nanda's user avatar
  • 15.4k
11 votes
2 answers
564 views

Local topology of Whitney stratified spaces

Let $M$ be a smooth manifold, let $\mathcal{P}$ be a Whitney stratification of $M$ and let $S\subset M$ be a stratum with closure $\overline{S}$. Question: Does there exist an open neighborhood $U\...
Jesse Wolfson's user avatar
3 votes
2 answers
380 views

Piecewise isomorphism versus equivalence in Grothendieck ring

$\DeclareMathOperator\Var{Var}$Let $K_{0}(\Var_{\mathbb{C}})$ be the Grothendieck ring of varieties over $\mathbb{C}$. The class of a variety, $X$, in $K_{0}$ is denoted $[\,X\,]$. If $X$ and $Y$ are ...
user avatar
1 vote
0 answers
28 views

Stratification which makes the defining functions isotrivial

Let $0\in X\subset\mathbb{C}^N$ be a germ of complex space and $0\in Z\subset X$ be a closed analytic subset (globally) defined by holomorphic functions $f_1,\dots,f_r$. Is there a complex analytic ...
stjc's user avatar
  • 1,072
5 votes
0 answers
77 views

subanalytic realization of smooth abstract stratification

Consider an $C^\infty$ abstract stratification $A$ (in the Thom-Mather sense, see Mather's note). Can we embed $A$ in some $\mathbb{R}^n$ (or in an analytic manifold) as a subanalytic set? If not, ...
Quentin's user avatar
  • 83
7 votes
1 answer
336 views

Non-example for Whitney (a) stratifications

Given a $C^1$ stratification $\mathscr{S}$ of a $C^1$ manifold $M$, we write $N^\ast \mathscr{S}$ for the union of conormals to the strata. The stratification is said to be Whitney (a) if $N^\ast \...
Chris Kuo's user avatar
  • 525
3 votes
0 answers
195 views

Genus two curves on abelian surfaces

Considering a smooth genus two curve $C_2$, let $J(C_2)$ be its Jacobian surface, and take $p \in J(C_2)$ an $m$-torsion point. Let $A = J(C_2)/Z_m$, where $Z_m$ acts by $x \mapsto x+p$. The image of $...
Rodion N. Déev's user avatar
6 votes
1 answer
408 views

Whitney stratification of algebraic varieties

When do the orbits of an action on an algebraic variety make a Whitney stratification?
Maicom Douglas Varella Costa's user avatar
8 votes
0 answers
244 views

Why mu-stratifications?

In the microlocal theory of sheaves developed by Kashiwara and Schapira, there is the notion of a $\mu$-stratification, which is a stratification satisfying a stronger property ("$\mu$") than Whitney'...
John Pardon's user avatar
  • 18.3k
4 votes
1 answer
119 views

Image of a quiver variety under natural morphism

We know that the natural morphism $\pi:\mathfrak{M}_{\theta}(Q,\mathbf{v},\mathbf{w})\rightarrow \mathfrak{M}_0(Q,\mathbf{v},\mathbf{w})$ between a smooth and affine quiver variety is not necessarily ...
Filip's user avatar
  • 1,617
2 votes
0 answers
177 views

Grothendieck group of constructible sets

Let $K_0$ be the Grothendieck group of complex algebraic varieties. This is the group generated by all complex algebraic varieties, subject to the relations: (i) $[X]=[Y]$ if $X,Y$ are isomorphic, (...
user142700's user avatar
4 votes
0 answers
382 views

A cell decomposition of a CW-complex and, stratification of a topological space

What is the difference between the notion of cell decomposition of a CW-complex, and the notion of stratification of a topological space ? I know that cell decomposition of a CW-complex is usefull to ...
YoYo's user avatar
  • 325
17 votes
0 answers
653 views

Proof of MacPherson's result about set-valued constructible sheaves and exit paths

I'm looking for a proof of a theorem that is attributed to MacPherson. Treumann (Section 1.1 in Exit paths and constructible stacks, 2009) states the theorem as: Theorem 1.2 (MacPherson). Let $(X,S)...
Jānis Lazovskis's user avatar
8 votes
0 answers
173 views

Stratification of space of labelled circles in the plane

Consider the space of $n$ round circles in the plane to be the open subset of $\mathbb R^{3n}$: $$C_n = \{ (v_1, v_2, \cdots, v_n, r_1, r_2, \cdots, r_n ) : v_i \in \mathbb R^2, r_i \in (0, \infty) \ ...
Ryan Budney's user avatar
  • 42.8k
1 vote
1 answer
135 views

Confusion about locally cone-like spaces

Definition: A filtered space $X$ of formal dimension $n$ is locally cone-like if for all $i$, $0 \le i \le n$, and for each $x \in X^i - X^{i-1} = X_i$ there is an open neighborhood $U$ of $x$ in $X_i$...
gf.c's user avatar
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1 vote
0 answers
197 views

Isn't stratification by orbit types actually a stratification by stibilizer types?

I asked this question on Math Exchange but considering the law number of people who viewed the question, I think that the question is difficult enough to post it on math overflow. I hope I am right. ...
Flavius Aetius's user avatar
18 votes
1 answer
587 views

Local homology of a space of unitary matrices

Let $U(n)$ denote the unitary group (this is a manifold of dimension $n^2$). Let $$ {\cal D} \subset U(n) $$ denote the subspace of those matrices having a non-trivial $(+1)$-eigenspace. ...
John Klein's user avatar
  • 18.6k
3 votes
2 answers
674 views

Whitney Conditions vs Equisingularity

In studying singular spaces, it is often important to pick an appropriate stratification which encodes the singularity structure. One class of such stratifications are called "Whitney stratifications" ...
Aswin's user avatar
  • 1,063
3 votes
1 answer
267 views

On the notion of conelike stratified (cs-) space

The notion of cs-stratification of a topological space is apparently due to Siebenmann, see also the paper by N. Habegger and L. Saper in the paper "Intersection cohomology of cs-spaces and Zeeman's ...
asv's user avatar
  • 21.1k
3 votes
0 answers
404 views

Where should I look for computing the intersection homology of projective varieties?

I'm learning about intersection cohomology topologically through MacPherson's "New York Times Article". This is a very nice guide which gives a nice idea on how to use these methods for low-...
54321user's user avatar
  • 1,706
1 vote
0 answers
31 views

Sufficient conditions for a conormal vector to be regular for an orbit stratification

Let a complex reductive group $G$ act on a $\mathbb{C}^{n}$ with finitely many orbits. Let $\mathcal{S}$ be the stratification of $\mathbb{C}^{n}$ according to these orbits. Let $(x,\xi) \in T_S^{*}\...
James Mracek's user avatar
11 votes
2 answers
404 views

Homotopy property of constructible sheaves on stratified spaces

Let $X$ be a stratified topological space (in my case $X$ is a compact space presented as a finite union of locally closed topological manifolds of finite dimension (strata) such that the closure of ...
asv's user avatar
  • 21.1k
8 votes
1 answer
507 views

Topology on the space of constructible sheaves

Let $X$ be a nice compact topological space with a fixed finite stratification by locally closed topological manifolds. At the beginning one may assume that $X$ is a complex algebraic manifold with ...
asv's user avatar
  • 21.1k
6 votes
1 answer
2k views

Stratification of complex algebraic varieties

Let $V$ be a complex quasi-projective variety, we know from H. Whitney's and B Teissier works on stratifications of algebraic varieties that $V$ has an intrinsic stratification $$X_0\subset X_2\...
David C's user avatar
  • 9,792
5 votes
0 answers
331 views

Stratification of a smooth map

So, this is an exercise. But from math.stackexchange I have been suggested to post this question here. To find the Thom-Boardman stratification of the smooth map $f(x,y,a,b,c,d)=x^2y+y^3+a(x^2+y^2)+...
PepeToro's user avatar
  • 231
7 votes
1 answer
559 views

Iterated Milnor fibrations and Thom's a_f condition

Ok so there's a lot of litterature about nearby cycles functor since it was introduced by Grothendieck and Deligne but I couldn't find any clear answer to the following natural question: Problem: Let ...
AFK's user avatar
  • 7,377
3 votes
2 answers
352 views

intersection of Whitney stratifications

Let $X$ be an oriented smooth manifold with dimension $n$. If $U$ and $V$ are two oriented closed submanifolds of $X$ and $U$ is transverse to $V$ in $X$. Then $U\cap V$ (suppose the intersection is ...
yangyang's user avatar
  • 237
0 votes
0 answers
223 views

transverse intersection of Whitney stratifications

Let $M$ be a smooth manifold. If $X$ and $Y$ are two Whitney objects, i.e. subsets with a given Whitney stratification, then $X$ and $Y$ are transverse if each stratum of $X$ is transverse to each ...
yangyang's user avatar
  • 237
2 votes
0 answers
186 views

When a Whitney stratification has no stratum of codimension one?

Let $G$ be a compact Lie group, and $M$ be a smooth $n$-dimensional $G$-manifold which admits an orientation preserving the $G$-action. Then $M$ has a natural Whitney stratification induced by the ...
yangyang's user avatar
  • 237
5 votes
0 answers
615 views

singular support of D-module smooth w.r.t. a stratification

(1) Suppose that $X$ is a smooth complex algebraic variety, stratified by some nice smooth stratification $S$. Let $M$ be a $D$-module on $X$, s.t. its shriek-pullback (or star... whatever is ...
Sasha's user avatar
  • 5,492
5 votes
1 answer
459 views

Is the Alexander-Pontryagin duality applicable to stratified spaces

If $D$ is the discriminant of the space of all planar curves of a fixed degree, and $D'$ is the subspace whose only singularities are nodes or cusps, then is it possible to apply Alexander-Pontryagin ...
user1289492's user avatar
6 votes
2 answers
1k views

Stratified pseudomanifold

In the definition of an $n$-dimensional stratified pseudomanifold one demands the following filtration $X=X_n \supset X_{n-1}=X_{n-2} \supset X_{n-3}\supset ... \supset X_0 \supset X_{-1}=\emptyset$. ...
Levi's user avatar
  • 63