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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
Fundamental group of the complement of a codimension two submanifold
To your first question, the answer is yes.
Take a $k$-component trivial link in $S^n$, i.e. the boring, linear embedding
$$\sqcup_k S^{n-2} \to S^n$$
that is the boundary of a linear embedding
$$\sqcu …
- 44.4k
4
votes
Lipschitz bounds and homotopy groups of diffeomorphism groups
I believe the answer is no, at least for the manifold $M = S^1 \times D^{m-1}$, when $m \geq 4$. At present I know it to fail for $k=0$ and $k=m-4$, but it likely fails for a broad range of values o …
- 44.4k
7
votes
Accepted
Can a nontrivial $n$-sphere bundle over $M$ embed in $M\times \mathbb{R}^{n+1}$?
A variation of your question has a positive answer. If you take any compact manifold that is a smooth bundle over another compact manifold $\pi : M \to N$, there is a smooth embedding
$$f : M \to N \t …
- 44.4k
11
votes
Accepted
Homotopy groups of the space of diffeomorphisms
There is no stability of the sort you are looking for. The reason is fairly simple-minded. For example, the orthogonal groups do not have the homotopic stability you are looking for, and diffeomorph …
- 44.4k
12
votes
Accepted
Transitivity of automorphism group of smooth manifolds
Partially this is a response to Mariano's 2nd comment.
In the smooth manifold case there's actually a really slick proof. Here it is:
Let $\gamma : [0,1] \to M$ be a smooth path in $M$ such that $\ga …
- 47.3k
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
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, potential …
- 44.4k
14
votes
Accepted
Isotopic diffeomorphisms of the sphere
This is Cerf's pseudoisotopy-implies-isotopy theorem.
Cerf's result is true in high dimensions, while it's independently known in a low-dimensional range. In dimension $2$ it goes back to the Earle–E …
- 82.7k
89
votes
Accepted
Can every manifold be given an analytic structure?
(similar to Mariano's post)
Q1: no. There are topological manifolds that don't admit triangulations, let alone smooth structures. All smooth manifolds admit triangulations, this is a theorem of White …
- 4,713
4
votes
Accepted
Does every simply connected, orientable, non-compact, 3-manifold embed in $\mathbb{R}^3$?
When the manifold is the universal cover of a compact $3$-manifold $M$ (to begin with, lets say without boundary) then you construct the embedding by hands, using geometrization. In your question let …
- 44.4k
6
votes
Stratification of smooth maps from R^n to R?
A standard reference is:
F. Sergeraert "Un theoreme de fonctions implicites sur certains espaces de Frechet et quelques applications," Ann. Sci. Ecole Norm. Sup. (4) 5 (1972), 599-660.
This isn't a st …
- 2,821
13
votes
Accepted
First cohomology of the space of long knots in ℝ⁴
Long knots in $\mathbb R^4$ form a simply-connected space. I pointed this out in my survey paper A Family of Embedding Spaces. The primary tool used to prove it is what's called the embedding calcu …
- 12.9k
15
votes
Accepted
Diffeomorphism group of the projective plane
Two different answers using almost identical techniques! Allen's response got me to think through my response more carefully. Let me edit in a comment to point out my sloppiness, as it points out a …
- 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
7
votes
The purpose of connections in differential geometry
On your more general question about differential geometry, i.e. why do people study it? There are many answers, some having little to do with each other.
In my opinion differential geometry is perha …
- 44.4k