10 votes
Accepted

Local homology of a space of unitary matrices

Let us first consider the case when $g=e$ is the identity matrix. Let $U$ be an open neighbourhood of the identity in $\mathcal D$. We want to calculate the local homology of $U$ at $e$. We may ...
Gregory Arone's user avatar
8 votes

Topology on the space of constructible sheaves

If you triangulate your space refining the stratification, a constructible sheaf is given by the data of a vector space $V_{\sigma}$ (a stalk at the barycenter, say) on each simplex $\sigma$ and a ...
David Treumann's user avatar
4 votes

Piecewise isomorphism versus equivalence in Grothendieck ring

After a little digging in the literature, I found the following example: Theorem. [KS18, Thm. 1.9] There exist non-isomorphic K3 surfaces $X$ and $Y$ over $\mathbf C$ such that $$[X \times \mathbf A^1]...
R. van Dobben de Bruyn's user avatar
4 votes
Accepted

Piecewise isomorphism versus equivalence in Grothendieck ring

There are no simple examples as yet; it's been an open question going back to at least Larsen and Lunts - Motivic measures and stable birational geometry, which has been open for about 15 years, and ...
Evgeny Shinder's user avatar
4 votes

Homotopy property of constructible sheaves on stratified spaces

Here are two comments: 1) I suspect the answer is "yes," so long as your homotopy has the property that your pullback is locally constant (and hence, by triviality of the interval, constant) along ...
Justin Curry's user avatar
  • 2,684
3 votes
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Whitney Conditions vs Equisingularity

I am not sure what you mean by "equisingular" stratification. But I guess you would like to say that $X$ is equisingular along $Y$ if the local rings $\mathcal{O}_{X,x}$ have constant multiplicity for ...
Libli's user avatar
  • 7,210
3 votes
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On the notion of conelike stratified (cs-) space

You should look at Greg Friedman's book: http://faculty.tcu.edu/gfriedman/IHbook.pdf CS-sets are discussed in section 2.3. In fact for many purposes it is rather interesting to suppose that the links $...
David C's user avatar
  • 9,792
3 votes

Non-example for Whitney (a) stratifications

I do not know of a simpler concrete example (as in the case of Whitney (b) condition) of a non-example for Whitney (a). But, a typical non Whitney (a) is as depicted in the picture. Observe that $X \...
Saurabh T's user avatar
  • 277
3 votes

Whitney stratification of algebraic varieties

The answer is indeed always when there are fnitely many orbits. Let $Y \subset X$ be an orbit, $y \in Y$, and $U \subset X$ a $G$-invariant open such that $y \in Y \cap U$, the relative open $Y \cap U$...
Libli's user avatar
  • 7,210
3 votes

Local topology of Whitney stratified spaces

In a paper written with a collaborator that we have recently uploaded on the arxiv, we show that indeed the conical charts of a Whitney stratified space provided by Thom and Mather induce a conically ...
mfox's user avatar
  • 283
3 votes
Accepted

Local topology of Whitney stratified spaces

In fact, any Whitney stratified set admits a stratification in the sense of Thom/Mather, cf Mather's notes on topological stability, published in B.A.M.S Volume 49, Number 4, October 2012, Pages 475–...
David C's user avatar
  • 9,792
3 votes

How to chart tubes around manifolds with boundary/corners?

From the comments I think the theorem you are looking for is this. I'll be a little fast and loose just to make it easier to state. Let $M$ be a manifold with corners and $N$ a submanifold, ...
Ryan Budney's user avatar
  • 42.8k
2 votes
Accepted

Smooth extension of piecewise smooth function on a corner

You have a function defined on the boundary of $\mathbb R^n_{\ge 0}$. Its restriction to each face is smooth. You can extend it to all of $\mathbb R^n_{\ge 0}$ by writing $$ f(x)=\sum_P (-1)^{|P|-1}...
Tom Goodwillie's user avatar
1 vote

Exit path categories of regular CW complexes

It seems to me like this statement is folklore, since e.g. the paper Stellar Stratifications on Classifying Spaces tries to show a generalization of it and at least hints that my simpler claim is true ...
Markus Zetto's user avatar
1 vote
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Comparing the exit path category and the nerve of a stratified space

This is false if the stratification is bad. (I first posted a more difficult example, but edited for simplification.) Example. Let $X = \mathbf R$, let $Z$ be the closure of $\big\{\tfrac{1}{n}\ \big|\...
R. van Dobben de Bruyn's user avatar
1 vote

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

The zero-dimensional stratum is the quotient of $\mathrm{Conf}_n(\mathbb{R})$ under the action of the group $\mathbb{R}_+ \rtimes \mathbb{R}$ of positive rescalings and translations. So for example ...
Najib Idrissi's user avatar
1 vote

Seeking a Weyl tube formula for Whitney stratified spaces

In the general case of algebraic sets (not necessarily complete intersections), you can take a look at this paper https://arxiv.org/abs/2104.05053 that extends Lotz' work.
A. Lerario's user avatar
1 vote

Whitney Conditions vs Equisingularity

Here is a result of Trotman relevant to your question. $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\ra}{\rightarrow}$ $\DeclareMathOperator{\cl}{cl}$ $\DeclareMathOperator{\dist}{dist}$ Suppose ...
Liviu Nicolaescu's user avatar
1 vote

Image of a quiver variety under natural morphism

The answer is completely known for ADE quiver varieties: Holds for general quiver varieties, as $\pi$ is a projective morphism so its image is a closed Poisson subvariety of $\mathfrak{M}_0$ so it ...
Filip's user avatar
  • 1,617
1 vote
Accepted

Confusion about locally cone-like spaces

Just to confirm your self-answer in the comments: That's right. In this case if you're looking for a neighborhood of $v$ in $X_0=X^0=\{v,w\}$, then $U=\{v\}$ does it and the cone neighborhood $N$ in $\...
Greg Friedman's user avatar
1 vote
Accepted

Homotopy property of constructible sheaves on stratified spaces

The above question has positive answer. It immediately follows from the following slightly more general statement. Proposition. Let $X$ be a compact topological space. Let $\mathcal{F}^\bullet\in D^b(...
asv's user avatar
  • 21.1k

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