Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with differential-topology
Search options answers only
not deleted
user 1345
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”.
47
votes
Parallelizability of the Milnor's exotic spheres in dimension 7
The tangent bundle to a smooth structure on $S^7$ is classified by a map $S^7 \to G_7(R^{\infty})$. By the exact sequence for a fibration for the fiber bundle $O(7)\to V_7(R^\infty)\to G_7(R^\infty)$, …
- 68.9k
43
votes
Is differential topology a dying field?
I don't think differential topology is a dying field.
I'll interpret this as the classification of smooth
manifolds and, more broadly, maps between them
(immersions, embeddings, diffeomorphism group …
- 68.9k
37
votes
What makes four dimensions special?
A comment is that 4 is the first dimension
for which every finitely presented group may be realized as the fundamental
group of a closed smooth 4-manifold. Other special properties are that the
first …
35
votes
Level sets of Morse functions
No such collection exists for $n=2$. This follows the construction in my paper "Small 3-manifolds of large genus".
The result of the paper is that for any $g$, there are closed orientable irreducible …
- 68.9k
25
votes
Accepted
Can one hear the (topological) shape of a drum?
There are examples due to Ikeda of isospectral Lens spaces which are not homotopy equivalent.
Likeliest the simplest examples are the compact connected 3-dimensional flat manifolds which are a tetra …
- 68.9k
18
votes
Accepted
Very particular kind of 4-manifolds. Classification
There are plenty of such manifolds, but as Danny indicates in his answer, there is not a known classification.
Take any acyclic group $G$ with a finite aspherical 2-complex $C$ with $\pi_1(C)=G$. Then …
- 68.9k
15
votes
Any 3-manifold can be realized as the boundary of a 4-manifold
The first part of question (4) makes sense, and the answer is yes by a
theorem of Freedman, who showed that any homology 3-sphere bounds a contractible 4-manifold. Applying this to the Poincaré homol …
- 68.9k
14
votes
Can you flip the end of a large exotic $\mathbb{R}^4$
As stated, your question is equivalent to the existence of a large exotic 4-ball (a smooth $D^4$ which cannot be smoothly embedded into $\mathbb{R}^4_{std}$).
The existence of a flip would give rise …
- 68.9k
13
votes
Self-covering spaces
A discussion (and partial classification with stronger assumptions on the cover) is given in a paper of van Limbeek. As Neil Hoffman points
out, a necessary condition is that the fundamental group is …
- 68.9k
12
votes
Accepted
A riemannian manifold with finitely many closed contractible geodesics
I think if you take the metric on $\mathbb{R}^2$ obtained by rotating a curve which is $\sqrt{1-x^2}$ for $-1\leq x\leq 0$, and $x^2+1$ for $x\geq 0$ around the $x$-axis, then I think there will be a …
- 68.9k
11
votes
Topological obstructions to existence of immersion
This is regarding (c) in the 2-dimensional case.
First, let's consider the orientable case.
Every orientable (connected) noncompact surface is a covering space of a genus 2 surface, and hence has a …
- 68.9k
10
votes
Isotopy in 3-manifolds
If $\Sigma_1 \hookrightarrow M$ is an embedded $\pi_1$-injective surface, then any homotopic embedded surface will be isotopic to $\Sigma$. As Ryan and Allen point out, this is due to Waldhausen for i …
- 68.9k
10
votes
Accepted
Representability of the sum of homology classes
There are a few well-known (at least to low-dimensional topologists) cases of this.
First, the obvious observation that if $k<d/2$, then one may take $Z_1\cap Z_2=\emptyset$ by a small isotopy, sinc …
- 68.9k
9
votes
On the generalized Gauss-Bonnet theorem
I'll make some general remarks. One may break down the generalized Gauss-Bonnet into two parts.
The first is to verify that it is a topological invariant. This means that
it should be independent of t …
- 68.9k
9
votes
Accepted
General position for map from surface to 3-manifold
In general, no you cannot. Consider one dimension down. Take two curves on a torus, intersecting transversely in a point. One of the curves cannot be homotoped to be disjoint from the pair of curves. …
- 68.9k