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Results tagged with differential-topology
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user 1465
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”.
4
votes
Accepted
Extending diffeomorphisms of some faces of the standard $n$- simplex to a diffeomorphism of ...
The answer to your specific question is no. There are some simple technical reasons for why there is a trivial "no" answer, but even if you deal with those there are more substantial reasons why the …
- 44.4k
8
votes
A kind of uniqueness for the double of a manifold
The answer is still no. I mis-read your question the first time. Finding a counter-example to your actual question was a little more difficult. I will produce a manifold that has two fundamentally …
- 44.4k
5
votes
Accepted
Examples of sphere bundles
As far as I know the only explicitly-described such bundles are in Hatcher's paper:
Hatcher. Concordance spaces, higher simple-homotopy theory, and applications. Algebraic and geometric topology ( …
- 44.4k
4
votes
Classifying manifolds
I suppose there's several closely-related reasons. One would be that manifolds are homogeneous objects -- they look the same near any point in them. So there's no natural thing to start counting if …
12
votes
Smooth representatives for elements of $\pi_7(\text{exotic $S^7$})$
I'll answer your question in two steps. (1) You can make a degree one map $f : S^7 \to M$ a homeomorphism, and $C^\infty$-smooth on the complement of a point.
The idea is that you can construct $M …
- 44.4k
5
votes
When is a manifold a tangent bundle?
One necessary set of conditions:
Step 1: you need $M$ to have the homotopy-type of a submanifold $N$ half the dimension of $M$. This forces $M$ to be a vector bundle over $N$ with the fibre the same …
- 44.4k
8
votes
Accepted
Neat maps between manifolds with boundary
Regarding your question 2, yes all maps of pairs $(Y,\partial Y) \to (X, \partial X)$ are homotopic to neat maps. There are many ways to prove it but it boils down to a collar construction. I would …
- 44.4k
8
votes
Isotopies of codimension-1 disks relative to boundary
This is a little different than the Schoenflies problem.
You can rephrase your question to be about the space of embeddings
$$D^{n-1} \to S^1 \times D^{n-1}$$
that agree with the standard embedding $\ …
- 44.4k
12
votes
Is the space of immersions of $S^n$ into $\mathbb R^{n+1}$ simply connected?
Let me just fill-in the gap in j.c.'s exposition. Smale-Hirsch states that the derivative from the space of immersions $Imm(S^n, \mathbb R^{n+1})$ to the space of bundle monomorphisms $Mono(TS^n, T\ma …
- 44.4k
8
votes
Accepted
Isotopy in 3-manifolds
No, generally they're not.
For example, there's only one homotopy class $S^2 \to \mathbb R^3$ but there's two isotopy classes of embeddings (given via how the embedding orients the compact 3-manifol …
- 44.4k
4
votes
Is there a Morse theory for sections of bundles or more generally for maps?
Extending Petya's answer, if you want something like indices, handle attachments and things of that flavour the answer is no. You can prove there's no such thing in that level of generality. But that …
- 44.4k
2
votes
Open map D⁴ → S²
This is too long for a comment but maybe it'll help me clarify what you're looking for. Interpret $D^4$ as the unit compact ball in $\mathbb C^2$.
$$ D^4 = \{ (z_1,z_2) \in \mathbb C^2 : |z_1|^2+| …
- 44.4k
1
vote
applications of Sard's to differential topology
It's been a long time, but isn't your suggestion roughly Whitney's original approach to this problem?
I don't have Whitney's papers in front of me but this is roughly how I think his arguments went. …
- 44.4k
19
votes
explicit diffeomorphim between open simplex and open ball
If the compact simplex is
$$\Delta_n = \{ (x_0,\cdots,x_n) : x_i \geq 0, x_0+x_1+\cdots+x_n=1\} \subset \mathbb R^{n+1}$$
then consider this function $f : \Delta \to \mathbb R \cup \{\infty\}$ def …
- 44.4k
9
votes
Extending a diffeomorphism of the sphere $S^2$ to the ball $D^3$
Regarding (2), Smale's proof can be made algorithmic. Smale's proof gives a formula for the extension. The formula involves some bump functions, the derivative, a lift of a derivative to the univer …
- 44.4k