# Questions tagged [compact-manifolds]

For questions regarding the structure and properties of compact manifolds.

214 questions
Filter by
Sorted by
Tagged with
53 views

### Homotopy invariance for compactly supported cohomology

I can't find any reference regarding homotopy invariance for compactly supported cohomology and I wonder under which conditions the homotopy invariance still holds for compactly supported cohomology. ...
40 views

### Why is this deformation retract well-defined?

I am trying to understand this proof from a paper, but I don't completely understand the deformation retract constructed in the last paragraph. For this deformation retract to be well-defined, we need ...
75 views

### Uniformization theorem for $C^k$ surfaces?

Does the uniformization theorem apply for surfaces that are $C^k$ ($k<\infty$)? I'm familiar with a couple of proofs of Uniformization (using Riemann-Roch, Ricci flow). But most of these proofs ...
21 views

### Is there an example of reducible compact 3-manifold with boundary that has no embedded incompressible two-sided surface?

There is a theorem stating that for irreducible compact manifolds with non-empty boundary there always exists such an embedded surface and I'm trying to understand why the irreducibility condition ...
51 views

28 views

### Compactification of log z Riemann Surface

I've been reading the 'Road to Reality' book of Roger Penrose and in the chapter on Riemann Surfaces, there is a note that we can compactify the log z Riemann Surface into a sphere. But I don't see ...
155 views

### What is the “natural homomorphism” in the definition of an *essential manifold*?

The following definition of "essential manifold" is in this wiki page: A closed $n$-manifold $M$ is called essential if its fundamental class $[M]$ defines a nonzero element in the homology ...
43 views

### Simplicial complexes embedded on a compact manifold

Every finite graph can be embedded on some compact surface of sufficient genus such that no two edges cross. If $S$ is a finite simplicial complex of dimension $n$, can $S$ be embedded in some ...
31 views

### Can all connected graphs be embedded on a closed, compact 2-Manifold?

I know that there are spherical (planar) graphs such as $K_4$,and toroidal graphs such as $k_7$, but I was wondering if given any connected graph $G$, there exists a closed, compact 2-Manifold $M$ ...
34 views

### An orientable surface that cannot be embedded into $\Bbb R^3$?

By Whitney's embedding theorem, every 2-dimensional manifold (aka. a surface) can be embedded into $\Bbb R^4$. Now, the Wikipedia page on that theorem states in this paragraph that we can even embedd ...
25 views

### intersection number on the boundary of a manifold

Let $F: W \to N$ be a smooth map, where $W$ is a compact manifold with boundary, $Z \subset N$ is closed and all manifolds are oriented. Also $\partial F \pitchfork Z$ and $F^{-1}(Z)$ is a compact, ...
27 views

79 views

### variation of laplacian on a compact riemannian manifold

I have a problem with a part of the proof in the article "prescribing curvature, Kazdan-Warner" of Lemma 3.2. In this lemma is required to compute the variation of the Laplacian operator, the authors ...
44 views

163 views

### Confusion about the top homology group of a compact manifold.

I know that if the manifold is compact, then all of its homology groups are finitely generated. But on the other hand, we know (for example Hatcher 3.26) that if the manifold is closed and orientable, ...
71 views

### $\mathbb{R}P^n$ can't be submanifold of $\mathbb{R}^n$

I need to prove that can't exist a function $f:\mathbb{R}P^n \rightarrow \mathbb{R}^n$ such that $(\mathbb{R}P^n,f)$ is a submanifold of $\mathbb{R}^n$. I can prove that for the case of $n$ even ...
137 views

### Is there any way to prove that the following space is a $2$-dimensional embedding manifold in $\mathbb{R}^3$?

How I can prove that: $X := \{(x, y, z) ∈ \mathbb{R}^3 \mid x^3 + y^3 + z^3 − 3xyz =1\}$ is a embedded manifold that $2$-dimensional in $\mathbb{R}^3$?
251 views

### 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) ...
38 views

### Reference SUBMANIFOLDS OF R^N

Is there any good reference that makes the theory of manifolds in $\mathbb{R}^N$. Especially area, coarea formulae, curvatures for submanifolds of $\mathbb{R}^N$, Differentiability, Vector fields, Lie ...
80 views

### Fiber bundle and local trivial fibration

If we have a fiber bundle $\pi:M\longrightarrow N$ which M and N be two compact smooth manifolds,is there a fibration on $M$? If we have a surjective submersion $\pi:M\longrightarrow N$ such that ...
32 views

### Is a compact connected manifold-with-boundary a CW complex?

Suppose $M$ is a compact connected manifold-with-boundary with non-empty boundary. What can be said on whether $M$ can be given a CW complex structure? A similar problem has been discussed for ...
53 views

### Is this proof that $M$ is orientable correct?

The exercise is the following: Let $M$ be a closed and connected topological manifold of dimension $n \geq 2$. If $H^1(M;\mathbb{Z}_2) = 0$, then $M$ is orientable. These are my thoughts: by ...
52 views

### Is the definition of the Sobolev space H^1(M) on a compact manifold that simple?

In https://hebey.u-cergy.fr/NotesSharpSP.pdf right at the beginning Hebey says Given $(M,g)$ a smooth compact $n$-dimensional Riemannian manifold, one easily defines the Sobolev spaces $H^p_k(M)$,...
87 views

### How are Sobolev spaces on compact Riemannian manifolds defined?

For an open subset $\Omega\subset \mathbb R^n$ one can define the Sobolev space $$H^1(\Omega):=W^{1,2}(\Omega)=\{u \in L^2(\Omega) \, \vert \, \partial u \in L^2(\Omega)\}.$$ Is there a "simple" way ...
60 views

### Projective bundle over complex algebraic variety

Let $X$ be a compact manifold which admits an embedding in the projective space and let $\pi: Y \to X$ a projective bundle on it. I'm trying to prove that also $Y$ is algebraic (embeddable in ...
32 views

### $H^1(\Omega)$ in Euclidean space vs $H^1(\Sigma)$ on a compact surface

For an open subset $\Omega \subset \mathbb R^n$ the Sobolev space $H^1(\Omega)=W^{1,2}(\Omega)$ is defined as \begin{equation} H^1(\Omega)=\{ u \in L^2(\Omega) \, \vert \, \partial^{\alpha}u \in L^2(\...
57 views

### Improve Lie algebra structure constant formula $f^{abc} f^{ade} \propto \delta^{b,d}\delta^{c,e}-\delta^{b,e}\delta^{c,d} + …?$

This is really a simple naive question. We know Levi-Civita symbol $\epsilon^{abc}$ has a nice property: https://en.wikipedia.org/wiki/Levi-Civita_symbol#Proofs  \epsilon^{abc} \epsilon^{ade}=\...
15 views

### Sobolev spaces on domains and manifolds, what is the difference?

What are the (technical) differences between Sobolev spaces on domains $\Omega \subset \mathbb R^n$ or (compact) manifolds such as two-dimensional spheres?
Why is the second Stiefel-Whitney Class of a closed oriented 4-manfifold, $M^{4}$, a characteristic element for its intersection form? Precisely, why must the following identity hold for closed ...
Let $M = \mathbb{R}\mathbb{P}^2 \times \mathbb{S}^1$. It is a prime 3-manifold, but it cannot be reducible, since the only reducible prime connected 3-manifolds are the $\mathbb{S}^2$-bundles over \$\...