All Questions
Tagged with singularity-theory differential-topology
12 questions
0
votes
0
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53
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How do I get f' from f that are A-equivalent?
I'm struggling with following problem realted to the singularity theory. I assume that $f, f' \in C^\infty(X,Y)$ and $f$ is a stable mapping. I would like to move from $f$ to some $f'$ which is $A$-...
- 1
2
votes
0
answers
83
views
A smooth map that is singular everywhere
Let $U$ be an open subset of $\mathbb{R}^d$, and let $F \colon U \to \mathbb{R}^{d +c}$ be a smooth map whose rank is everywhere equal to $d-1$:
$$\mathrm{rank} (dF_{p}) = d-1 \quad \text{for all $p \...
user242708
2
votes
0
answers
81
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calculate the kernel of a germ map
I have a question, more about how to calculate the kernel of a certain map:
Let the ring $\varepsilon_n=\lbrace f\colon (\mathbb{R}^{n},0)\longrightarrow \mathbb{R}: f \quad\textit{is map germ}\...
- 530
2
votes
1
answer
565
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Unknot: embedded vs immersed bounding disk
Suppose that we have a knot $K\subset \mathbb{S}^3$. If $K$ bounds* an embedded 2-disk then $K$ is the unknot.
But what happens if $K$ bounds an immersed 2-disk?
The immersed disk generically will ...
- 5,960
6
votes
2
answers
290
views
Can we approximate a vector field on the plane with non-vanishing vector fields in $L^2$?
Let $V$ be a compactly-supported smooth vector field on $\mathbb{R}^2$, whose zeros inside some open neighbourhood of the closed unit disk $\mathbb{D}^2$ are isolated.
Does there exist a sequence ...
- 25.6k
0
votes
1
answer
124
views
Same homotopy type as circle
Let $S=\{(z_1,z_2,\ldots, z_n)\in \mathbb{C}^n: |z_1|^2+|z_2|^2+\ldots+|z_n|^2=1\}$ let $K=\{(z_1,z_2,\ldots, z_n)\in S: |z_1|^2+|z_2|^2+\ldots+|z_{n-1}|^2=1\}$. Prove that $S\setminus K=M$ has same ...
- 969
3
votes
2
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78
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How can I see mathematically that these two singularities are different?
I came across these two curves while reading the wikipedia page on singularity theory:
$$
y^2=x^3+x^2
$$
and
$$
y^2=x^3
$$
The page says the cusp at $(0,0)$ can be seen to be qualitatively different ...
- 29.5k
1
vote
0
answers
447
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Simple singularities of a vector field
Let $M^n \subset \mathbb{R}^{n+1}$ a compact and orientable hypersurface of even dimension, with Gauss map $\gamma : M \to S^n$, that is, $\gamma(p)$ is normal to $M$ at $p \in M$. Let $a, b \in S^n$ ...
- 5,235
4
votes
1
answer
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Mathematical definition of the word "generic" as in "generic" singularity or "generic" map?
I've been trying to work out what generic means but I'm not making much progress.
You can find an example of the usage of the word generic for example here: "School on Generic Singularities in ...
- 1,647
6
votes
1
answer
524
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Topology of Isolated Singularities
In Singular Points of Complex Hypersurfaces, Milnor shows that if
$f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$, holomorphic, has an isolated
singularity at the origin, then $f^{-1}(\epsilon) \cap B^{...
- 1,554
8
votes
1
answer
812
views
Does every open manifold admit a function without critical point?
Assume that $M$ is a non compact smooth manifold. Is there a smooth map $f:M\to \mathbb{R}$ such that $f$ has no critical point?
The motivation comes from the conversations on this post.
- 851
25
votes
3
answers
8k
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How to know if a tangent bundle is trivial from its defining equations
In this question, I am considering only regular manifolds.
A Trivial Bundle
The circle $S^1$ is known to have a trivial tangent bundle.
As a subset of $\mathbb{R}^4$, the tangent bundle of $S^1$ ...
- 32.7k