All Questions
Tagged with singularity-theory general-topology
8 questions
6
votes
1
answer
604
views
Condition for polynomials to be proper
Let $\Bbbk\in \left\{ \mathbb R,\mathbb C \right\}$. Suppose $\mathbb \Bbbk^n\overset{f}{\to} \Bbbk$ is a homogeneous polynomial map satisfying the following condition: the fiber of $f$ containing the ...
- 14.2k
0
votes
0
answers
46
views
Singularities from topological viewpoint
I intuitively understand that
\begin{align}
f:(-\infty.0)\cup(0,+\infty) &\to \mathbb{R} \\
x &\mapsto \dfrac{1}{x}
\end{align}
is "singular" at $x=0$, however, is there a topological ...
- 1,350
1
vote
0
answers
76
views
Non-homeomorphic spaces that can have homeomorphic open cones.
Apparently such a space exists, and apparently Milnors solution to Hauptvermutung also explains this situation. I am learning about topologically stratified spaces, so it would be nice to know a ...
- 21.2k
11
votes
2
answers
1k
views
Do the singular matrices form a topological manifold
So the definition of manifold I'm using is that of a topological manifold (a topological space with an atlas of homeomorphisms to $\mathbb{R}^n$).
I have two related questions:
Is the set of ...
- 8,091
7
votes
1
answer
1k
views
Books or texts on singularity theory [closed]
So a friend is doing his PhD in maths (algebraic topology) and his advisor wants him to publish something on singularities (of which, as fas as I understand, he knows next to nothing). I want to give ...
- 79
0
votes
0
answers
62
views
The normalisation map is a bi-Lipschitz map?
Let $X$ a reduced analytic space, $n: W \rightarrow X$ the normalisation map, $W$ the normalisation of $X$ and $S$ the singular set of $X$. When we restrict $n$ to $W\setminus n^{-1}(S)$, we know that ...
- 33
1
vote
0
answers
248
views
Stratification of a smooth map
I am trying to do an exercise. Namely, find the Thom-Boardman stratification of the smooth map
$f(x,y,a,b,c,d)=x^2y+y^3+a(x^2+y^2)+bx+cy$, where $a,b,c$ are parameters. As I have seen, this is also ...
- 1,016
3
votes
2
answers
650
views
The notion of a germ in singularity theory
I quote from my lecture:
Let $X$ be a topological space (think of $X=\mathbb{C}^n$ with the classical topology), $p\in X$, $A,B\subseteq X$. Then $A\sim B$ if there exists an open subset $U\...
- 2,524