11
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 ...
- 10.9k
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]...
- 28.7k
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 ...
- 2,300
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 \...
- 287
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$...
- 7,300
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 ...
- 303
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–...
- 9,870
3
votes
Accepted
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 ...
- 7,300
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, ...
- 44.4k
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}...
- 55.9k
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 ...
- 898
1
vote
Accepted
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|\...
- 28.7k
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 ...
- 5,040
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.
- 263
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 ...
- 34.7k
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 ...
- 1,677
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 $\...
- 5,489
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