All Questions
Tagged with dg.differential-geometry differential-topology
759 questions
142
votes
17
answers
23k
views
What makes four dimensions special?
Do you know properties which distinguish four-dimensional spaces among the others?
What makes four-dimensional topological manifolds special?
What makes four-dimensional differentiable manifolds ...
113
votes
4
answers
13k
views
Is there a sheaf theoretical characterization of a differentiable manifold?
I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...
- 22.1k
94
votes
4
answers
15k
views
Can every manifold be given an analytic structure?
Let $M$ be a (real) manifold. Recall that an analytic structure on $M$ is an atlas such that all transition maps are real-analytic (and maximal with respect to this property). (There's also a sheafy ...
- 54.6k
84
votes
4
answers
6k
views
Parallelizability of the Milnor's exotic spheres in dimension 7
Are the Milnor's seven dimensional exotic spheres parallelizable?
- 1,236
78
votes
7
answers
8k
views
Example of a manifold which is not a homogeneous space of any Lie group
Every manifold that I ever met in a differential geometry class was a homogeneous space: spheres, tori, Grassmannians, flag manifolds, Stiefel manifolds, etc. What is an example of a connected smooth ...
- 8,559
69
votes
4
answers
13k
views
What is a foliation and why should I care?
The title says everything but while it is a little bit provocative let me elaborate a bit about my question. First time when I met the foliation it was just an isolated example in the differential ...
- 9,330
64
votes
1
answer
4k
views
A dictionary of Characteristic classes and obstructions
I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians.
In an effort to ...
- 7,789
63
votes
0
answers
2k
views
Are there periodicity phenomena in manifold topology with odd period?
The study of $n$-manifolds has some well-known periodicities in $n$ with period a power of $2$:
$n \bmod 2$ is important. Poincaré duality implies that odd-dimensional compact oriented manifolds ...
- 118k
59
votes
3
answers
5k
views
Operations via Morse Theory
I am interested in seeing if and how Morse Theory can "do everything". Some core things are handle decomposition, Bott periodicity, and Euler characteristic. But what do the normal (co)...
- 17.5k
41
votes
3
answers
3k
views
Can one recover the smooth Gauss Bonnet theorem from the combinatorial Gauss Bonnet theorem as an appropriate limit?
First let me state two known theorems.
Theorem 1 (for smooth manifolds): Let $(M,g)$ be a smooth compact two dimensional Riemannian manifold. Then
$$ \int \frac{K}{2 \pi} dA = \chi (M) $$
where $K$ ...
- 3,245
40
votes
3
answers
3k
views
Height function on 2-torus with only 3 critical points
It is well-known that a Morse function on $T^2$ has at least $4$ critical points, but also that there exist functions $f\colon T^2\to\mathbb R$ with only 3 critical points (the least possible number ...
- 5,891
39
votes
10
answers
4k
views
Are there some other notions of "curvature" which measure how space curves?
I am learning differential geometry and have a few questions on curvature. -- Background:
Gauss invented "Gauss curvature" to measure how surface curves.
Riemann gives an ingenious generalization of ...
39
votes
4
answers
9k
views
How to tackle the smooth Poincaré conjecture
The last remaining problem in this whole "everything is a sphere" business, is the smooth Poincaré conjecture in dimension 4: If $X\simeq_\text{homo.eq.} S^4$ then $X\approx_\text{diffeo} S^...
- 17.5k
39
votes
2
answers
9k
views
Exotic differentiable structures on R^4?
This was going to be a comment to Differentiable structures on R^3, but I thought it would be better asked as a separate question.
So, it's mentioned in the previous question that $\mathbb{R}^4$ has ...
- 2,179
36
votes
2
answers
5k
views
Kervaire invariant: Why dimension 126 especially difficult?
Is there any resource that might help non-experts gains some understanding of why
the Kervaire invariant problem remains open now only in dimension $126$? ($126 =2^7-2=2^{j+1}-2$;
whether $\theta_j=\...
- 151k
36
votes
1
answer
2k
views
Can a topological manifold have different tangent bundles?
We know that the tangent bundles of the sphere arising from different smooth structures are equivalent as vector bundles. Is it right in general? I want to know the relationship between the set of ...
- 1,799
34
votes
10
answers
11k
views
Why is cotangent more canonical than tangent?
You don't need a metric to define the differential of a function,
and the cotangent bundle carries a canonical one-form.
But you do need a metric to define the gradient, and the
tangent bundle does ...
- 3,267
34
votes
1
answer
4k
views
Strong Whitney embedding theorem for non-compact manifolds
$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds.
The strong ...
- 6,210
33
votes
4
answers
7k
views
Topology of function spaces?
Let $X,Y$ be finite-dimensional differentiable manifolds, and let's assume that they are connected. In fact, in applications I would like both $X$ and $Y$ to be riemannian manifolds.
Let $C^\infty(X,...
- 33.3k
32
votes
2
answers
2k
views
Converse to Stokes' Theorem
Does satisfying Stokes' Theorem imply that a form is linear?
Let $M$ be an $n$-manifold. A differential $k$-form $\omega \in \Omega^k M$ assigns to each point $x \in M$ a function $\omega_x : \Lambda^...
- 63.9k
31
votes
1
answer
1k
views
What results about the topology of manifolds depend on the dimension mod 3?
There are a lot of interesting results about the topology of manifolds that depend on the dimension of the manifold mod 2, mod 4, or mod 8. The simplest ones involve the cup product
$$ \smile \colon ...
- 22.3k
29
votes
2
answers
2k
views
A simple proof that parallelizable oriented closed manifolds are oriented boundaries?
So let $M$ be a smooth closed orientable real manifold such that $M$ is parallelizable, i.e., the tangent space $TM$ of $M$ is trivial. From the triviality of $TM$ we get that the Stiefel-Whitney and ...
- 7,609
29
votes
4
answers
3k
views
Conceptual proof of classification of surfaces?
Every compact surface is diffeomorphic to $S^2$, $\underbrace{T^2\#\ldots \#T^2}_n$, or $\underbrace{RP^2\#\ldots \#RP^2}_n$ for some $n\ge 1$.
Is there a conceptual proof of this classification ...
- 43.2k
29
votes
2
answers
2k
views
Contractibility of the space of Jordan curves
Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.
If the curves are ...
- 7,149
29
votes
3
answers
2k
views
Embeddings of $S^2$ in $\mathbb{CP}^2$
Suppose we are given an embedding of $S^2$ in $\mathbb{CP}^2$ with self-intersection 1. Is there a diffeomorphism of $\mathbb{CP}^2$ which takes the given sphere to a complex line?
Note: I suspect ...
- 6,247
29
votes
1
answer
4k
views
Smooth bijection between non-diffeomorphic smooth manifolds?
The "textbook" example of a smooth bijection between smooth manifolds that is not a diffeomorphism is the map $\mathbb{R} \rightarrow \mathbb{R}$ sending $x \mapsto x^3$. However, in this example, ...
- 2,713
29
votes
0
answers
2k
views
Nontrivial tangent bundle that is diffeomorphic to the trivial bundle
Is there an example of a smooth $n$-manifold $M$ whose tangent bundle is nontrivial as a bundle but is nonetheless (abstractly) diffeomorphic to the trivial bundle $M \times \mathbb{R}^n$?
(This ...
- 10.3k
26
votes
2
answers
2k
views
Euler characteristic and universal cover
Let $M$ be a compact manifold, let $\tilde{M}$ be its universal cover, and suppose that the Euler characteristic $\chi(\tilde{M})=0$.
My question is: does this imply that $\chi(M)=0$?
This is clear if ...
- 1,452
26
votes
2
answers
1k
views
Vector fields on $(4n+1)$-spheres
If $n$ is odd then $S^{n-1}$ doesn't admit a nowhere-vanishing vector field, and if $n$ is even then there does exist one (Hairy Ball Theorem). We can then ask, on $S^{n-1}$, what is the maximum ...
- 17.5k
25
votes
1
answer
4k
views
Can the constant rank theorem for smooth manifolds be generalized to nonconstant rank?
The constant rank theorem says that
if $f\colon M→N$ is a smooth map whose rank equals some fixed $k≥0$ at any point of $M$, then, locally with respect to $M$ and $N$, the map $f$ assumes the easiest ...
- 37.8k
24
votes
1
answer
1k
views
All fiber bundles over $S^2$ extendable to $\mathbb{C}P^\infty$?
I ran into the following sanity check. Is the following statement true?
Every smooth fiber bundle (with compact fiber) over $S^2$ can be extended to a smooth fiber bundle over $\mathbb{C}P^\infty$ (...
- 707
23
votes
1
answer
2k
views
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 $...
- 491
23
votes
2
answers
928
views
Why is it true that if two 4-manifolds are homeomorphic then their squares are diffeomorphic?
Near the top of the second page of this paper, it is claimed that if two 4-manifolds $X$ and $Y$ are homeomorphic, then their squares $X \times X$ and $Y \times Y$ are diffeomorphic. Why is this true? ...
- 375
22
votes
5
answers
5k
views
On the generalized Gauss-Bonnet theorem
I am trying to learn about basic characteristic classes and Generalized Gauss-Bonnet Theorem, and my main reference at the moment is From calculus to cohomology by Madsen & Tornehave. I know the ...
- 1,719
22
votes
2
answers
2k
views
Examples of loss of regularity by "creation of topology"
I would like to have a list as general as possible of examples of situations where the density of smooth objects into some "natural class" (the meaning of "natural" depending on the problem considered)...
- 2,041
21
votes
3
answers
2k
views
Manifolds with polynomial transition maps
Title edited I thank მამუკა ჯიბლაძე and Corbennick for their suggestion on the title of this question. I changed the title based on the suggestion of Corbennick.
What is an example of a ...
- 356
21
votes
1
answer
2k
views
Intuition behind manifolds which are homeomorphic but not diffeomorphic
Popular articles on mathematics often explain the difference between homeomorphism and diffeomorphism with statements like - "A rectangle is homeomorphic to the circle but not diffeomorphic to it&...
- 463
20
votes
4
answers
8k
views
What is an immersed submanifold?
An immersed submanifold is by definition the image of a smooth immersion. I know some examples but I lack general understanding of what immersed submanifolds look like. For example, can one ...
- 29.1k
20
votes
2
answers
2k
views
Can one hear the (topological) shape of a drum?
Let $(M,g)$ be a (say closed) Riemannian manifold. One can try to understand the geometry/topology of $(M,g)$ by studying the eigenvalues of the Laplacian (this I guess has two versions: when ...
- 953
20
votes
5
answers
2k
views
Smoothness of the closest point on a submanifold
Let $(M,g)$ be a smooth Riemannian manifold, and let $S \subseteq M$ be a compact submanifold.
Assume that for each $p \in M$, there exist a unique closest point on $S$, i.e a unique point $\tilde s(...
- 6,741
19
votes
8
answers
2k
views
Theorems that led to very successful research programs in Geometry and Topology [closed]
In the recent times I have heard a lot about the following:
The Atiyah-Singer Index theorem
H-principle of Gromov ( and others )
It seems to me that these results led to decades of successful ...
19
votes
3
answers
3k
views
When does the tangent bundle of a manifold admit a flat connection?
Let $M$ be a smooth manifold, and let $TM$ denote its tangent bundle. Under what conditions does $TM$ admit a flat connection $\omega$?
Edit: Formerly, I asked about a flat connection on the frame ...
- 8,512
19
votes
1
answer
989
views
Can the product of a 3-dimensional lens space with a circle be diffeomorphic to another such product when the lens spaces aren't diffeomorphic?
This is a question that I need to answer in order to resolve an issue for my dissertation and I am looking for a reference. Here is the precise statement of the question.
Suppose we have two three-...
- 293
19
votes
1
answer
765
views
Are Hölder manifolds a thing?
We know topological manifolds and we know Lipschitz manifolds. It seems that "Hölder manifolds" should be somewhere in between but not much seems published about them.
In the context of this question,...
- 5,327
19
votes
0
answers
312
views
Can one properly embed a differential manifold into numerical space of double dimension? [duplicate]
If $X$ is a $ C^\infty$ differential manifold of dimension $n$, then there exists an embedding $f:X\to \mathbb R^{2n+1}$.
This is a not too difficult theorem due to Whitney, proved in many textbooks.
...
- 47.5k
18
votes
4
answers
976
views
Is the space of immersions of $S^n$ into $\mathbb R^{n+1}$ simply connected?
The title is the question. Sorry, this isn't quite research level. I imagine the answer is well-known, just not to me. Thanks for any help!
18
votes
3
answers
4k
views
What is an "Instanton" in classical gauge theory? (to a mathematician)
There's already a question about the same topic but I think its aim is different.
Classical (non-quantum) gauge theory is a completely rigorous mathematical theory. It can be phrased in completely ...
- 7,789
18
votes
4
answers
3k
views
A topological consequence of Riemann-Roch in the almost complex case
This question originated from a conversation with Dmitry that took place here
Is there a complex structure on the 6-sphere?
The Hirzebruch-Riemann-Roch formula expresses the Euler characteristic of ...
- 23.5k
18
votes
1
answer
1k
views
Is the minimal volume a topological invariant?
On Wikipedia, it is said that the minimal volume
$$\operatorname{MinVol}(M):=\inf\{\operatorname{vol}(M,g) :g\text{ a complete Riemannian metric with }|K_{g}|\leq 1\}$$
is a topological invariant, ...
- 609
18
votes
1
answer
1k
views
Approximation of homeomorphism by diffeomorphism
Let $M$ be a smooth closed manifold. Let $f\colon M\to M$ be a homeomorphism.
Does there exist a sequence of diffeomorphisms $f_i\colon M\to M$ which conveges to $f$ uniformly, i.e. in $C^0$-...
- 21.8k