For questions regarding the structure and properties of compact manifolds.
Compact manifolds are manifolds which are compact as topological spaces.
Questions tagged [compact-manifolds]
252 questions
28
votes
1 answer
Understanding Takens' Embedding theorem
I am having some trouble understanding Takens' embedding theorem, and was hoping that someone with greater knowledge could help out.
Formally, the theorem goes as follows:
Let $M$ be a compact manifold of dimension $m$. For pairs $(\phi,y)$, where…
Astrid
- 742
20
votes
3 answers
Examples of manifolds that are not boundaries
What are some examples of manifolds that do not have boundaries and are not boundaries of higher dimensional manifolds?
Is any $n$-dimensional closed manifold a boundary of some $(n+1)$-dimensional manifold?
Xiaoyi Jing
- 497
- 4
- 12
18
votes
1 answer
If the connected sum of a manifold $M$ with itself gives back $M$, does it imply $M$ is a sphere?
Let $M$ be a compact, connected, oriented $n$-dimensional manifold without boundary. Suppose that $M\#M\cong M$. Does it imply that $M \cong S^n$?
Sorry if this is a naive question. This is not my area, and I have very few examples of higher…
Bruno Joyal
- 55,975
18
votes
1 answer
Does there exist a compact submanifold of $\mathbb{R}^3$ whose fundamental group is $\mathbb{Z}^3$?
Does there exist a compact submanifold of $\mathbb{R}^3$ whose fundamental group is $\mathbb{Z}^3$ ?
The question in the title is a generalization of the question that really interests me:
Does there exist a connected finite set of unit cubes of…
Arshak Aivazian
- 915
15
votes
2 answers
Why spheres are not symplectic manifolds?
Reading some books on diferential geometry, a found that $S^{2n}$ (with $ n>1$) are not symplectic manifolds. They say it's because the de Rham cohomology of this spheres are R, but I do not understand this argument. It will be helpfull if anyone…
Edgar Ortiz
- 373
14
votes
1 answer
Norm Inequality on a Compact Riemannian Manifold
Consider the following problem:
Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality
$$\Delta u \geq -cu$$
where $c$ is a constant.
Show…
user100463
13
votes
2 answers
A non orientable closed surface cannot be embedded into $\mathbb{R}^3$
Can someone please remind me how this goes?
Here's the idea of proof I'm trying to recall: let $S$ be a closed surface (connected, compact, without boundary) embedded in $\mathbb{R}^3$. Then one can define the "outward-pointing normal unit vector"…
Seub
- 6,282
13
votes
5 answers
Could exists a vector field on $\mathbb{S}^{2}$ with exactly $n$ zeroes?
I just started to learn index theory of tangent vector fields. I'm aware of two examples on the sphere $\mathbb{S}^{2}$ with exactly one zero, which, which are $F(x,y) = (1-x^2-y^2)\partial x$ thought on $\mathbb{D}^2$ and then identify the…
jacopoburelli
- 5,324
12
votes
1 answer
Geometric intuition for torsion in $H_{2}$ of non-orientable $3$-manifold
Let $M$ be a compact, connected $n$-manifold. Consider the homology groups $H_n(M)$ with coefficients in $\mathbb{Z}$.
It is well known that if $M$ is not $\mathbb{Z}$-orientable, then we have $H_n(M) =0$ and $H_{n-1}(M) = \mathbb{Z}/2 \oplus…
user267839
- 9,217
12
votes
2 answers
Finitely generated singular homology
Let G be a finitely generated abelian group and M a compact manifold, I want to prove that $H_r(M,G)$ is finitely generated for $r\ge 0 $.
First I was thinking if I could do induction over $r$ because for $r=0$, $H_0(M,G) \cong \bigoplus_{i=1}^nG$…
jandrew
- 591
- 6
- 12
11
votes
1 answer
Euler characteristic of a manifold is odd
This was a past exam question: Let $M$ be a compact connected orientable topological $n$-manifold with boundary $\partial M$ so that $H_*(\partial M;\mathbb{Q}) \cong H_*(S^{n-1};\mathbb{Q})$. If $n \equiv 2$ mod $4$, show that the Euler…
xy15
- 149
11
votes
1 answer
Affine manifolds which are not euclidean manifolds.
I want to find a differentiable $n$-dimensional compact manifold $M$ which can be endowed with an affine structure but cannot be endowed with a euclidean structure.
An affine (resp. euclidean) structure is a geometric structure with $X=\Bbb R^n$…
Adam Chalumeau
- 3,333
10
votes
0 answers
Homology of a compact manifold is finitely generated
Perhaps this is obvious and I am overlooking something, but why are the homology groups of compact manifolds finitely generated?
Holdsworth88
- 8,948
10
votes
3 answers
Dimension of de Rham Cohomology groups?
Is there a simple way to prove that the de Rham cohomology groups of a compact manifold $M$ have finite dimension as $\mathbb{R}$-vector spaces?
Heitor Fontana
- 3,756
8
votes
4 answers
1-manifold is orientable
I am trying to classify all compact 1-manifolds. I believe I can do it once I can show every 1-manifold is orientable. I have tried to show prove this a bunch of ways, but I can't get anywhere.
Please help,
Note, I am NOT assuming that I already…
Bates
- 940