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Results tagged with differential-topology
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user 89166
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
23
votes
1
answer
2k
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Is the normal bundle of a torus trivial?
Question:
Let $T^k \subseteq \mathbb{R}^n$, $ n > k$, be a smoothly embedded $k$-torus. Is its normal bundle trivial?
What about the normal bundle of $S^k \subseteq \mathbb{R}^n$, $n > k$, the $k$-sp …
- 501
8
votes
0
answers
200
views
Maps with small fibers between manifolds of equal dimension
The following question is an attempt to revise this one into what I intended.
Important revisions are shown in bold.
Are there any known examples of a compact Riemannian manifold $M$ with (possibly em …
- 501
8
votes
1
answer
302
views
Fibers of generic smooth maps between manifolds of equal dimension
I have heard that the following is a "well-known"
Claim. Let $M$ and $N$ be smooth manifolds with equal dimensions and $M$ compact. Then a generic smooth map $f\colon M\to N$ has finite fibers, i.e., …
- 501
7
votes
1
answer
205
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Recognizing sections up to isotopy
Let $E$, $B$ be smooth manifolds, $\pi\colon E\to B$ be a smooth fiber bundle, and $h:B\to E$ be a smooth embedding. I would like to learn what is known about the following
Question. When does there e …
- 501
6
votes
2
answers
687
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On Wilson's claim that Lyapunov function level sets are not exotic spheres
In Wilson's paper "The structure of the level surfaces of a Lyapunov function," he states in Corollary 1.3 that the level sets of a smooth Lyapunov function are diffeomorphic to a standard sphere. (Th …
- 501
5
votes
1
answer
404
views
Making a submanifold transverse to a vector field by an isotopy
Let $M$ be a smooth manifold, $N\subset M$ be a smooth closed hypersurface not bounding a compact submanifold, and $X$ be a smooth nowhere-zero vector field on $M$. I would like to learn what is known …
- 501
5
votes
1
answer
245
views
Codimension zero embeddings and maps with small fibers
Edit: as explained in my comment on alesia's answer, I mistakenly did not ask below the question I intended (due to my misguided efforts to simplify it). Thus, I revised and reposted my question here. …
- 501
4
votes
2
answers
759
views
Are there vector fields which are gradients with respect to one metric but not another? [closed]
Is it possible for a vector field on a smooth manifold $M$ to be a gradient with respect to a Riemannian metric $g$, but not a gradient with respect to a different Riemannian metric $h$?
For complete …
- 501
4
votes
1
answer
257
views
A cobordism theory from Hirsch's "Differential Topology" (reference request)
The following is exercise 5 on p. 176 in Hirsch's "Differential Topology" (corrected 6th printing):
Let $\eta = (p,E,B)$ be a fixed vector bundle over a compact manifold $B$, $\partial B = \varnothin …
- 501
4
votes
0
answers
111
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Do any Legendrian knots in standard contact 3-space have big tubular neighborhoods?
Consider $\mathbb{R}^3$ with the standard contact structure $\ker(dz-y\,dx)$.
According to the contact version of Weinstein's theorem, any Legendrian knot $L\subset \mathbb{R}^3$ has a tubular neighbo …
- 501
3
votes
0
answers
83
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Is the increasing union of disk bundles a disk bundle?
Setup: Let $B$ be a $C^r$ $n$-manifold ($r \geq 1$) and $M$ a closed $k$-dimensional $C^r$ submanifold of $B$. Assume there exists a smooth retraction $p:B \to M$ which is also a submersion, so that $ …
- 501
3
votes
0
answers
82
views
Are these contact structures on the open solid torus diffeomorphic?
Let $M=S^1\times \mathbb{R}^2$ and $\alpha_1, \alpha_2$ be a pair of contact one-forms on $M$ such that the restrictions $\alpha_1|_{S^1\times \{0\}}$, $\alpha_2|_{S^1\times \{0\}}$ coincide and satis …
- 501
3
votes
1
answer
188
views
Can a nontrivial $n$-sphere bundle over $M$ embed in $M\times \mathbb{R}^{n+1}$?
Let $\pi\colon E\to M$ be a smooth $S^n$-bundle with structure group $\text{Diff}(S^n)$.
Assume there is a smooth embedding $f:E\to M \times \mathbb{R}^{n+1}$ such that $\text{pr}_1 \circ f = \pi$, wh …
- 501
2
votes
0
answers
221
views
On "graphs" of foliations
Let $M$ be a smooth manifold and $\mathcal{F}=\{\mathcal{F}_m\}_{m\in M}$ be a (regular) smooth foliation of $M$. The leaves $\mathcal{F}_m$ are smoothly immersed and moreover weakly embedded submanif …
- 501
2
votes
0
answers
192
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Classification of bundles with fixed total space
I am aware of classification theorems for principal bundles, vector bundles, and covering spaces $\pi:E\to B$ over a fixed base space $B$. Principal and vector bundles over $B$ are classified by homot …
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