Questions tagged [compact-manifolds]

For questions regarding the structure and properties of compact manifolds.

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3
votes
2answers
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. ...
0
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0answers
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 ...
2
votes
1answer
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 ...
1
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0answers
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 ...
2
votes
1answer
51 views

Making intuition rigorous that integral of some positive function on set should be monotone in the Haar measure of the set

Let $\mathcal{M}$ be a compact Riemannian manifold with geodesic distance function $d$ and $\Omega$ its volume measure. Pick some $A,B\subseteq\mathcal{M}$ such that $\Omega(A)\ll\Omega(B)$, but: (1) $...
4
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1answer
55 views

Covering $\Bbb RP^\text{odd}\longrightarrow X$, what can be said about $X$?

I am looking for any argument related to the following fact, which may or may not be true. Let $f:\Bbb RP^n\longrightarrow X$ be a covering space, where $n\geq 2$. Then, $X=\Bbb RP^n$. Now, for $n=\...
0
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1answer
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 ...
4
votes
2answers
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 ...
0
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1answer
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 ...
0
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1answer
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 $M$ ...
4
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0answers
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 ...
0
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0answers
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, ...
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0answers
27 views

The Sobolev embedding inequality on manifolds

Let $(M,g)$ be a (smooth) compact Riemanian manifold of dimension $n$. I expect that the following inequality is true for any smooth function $f$: $$(\int_{M} |f|^{\beta})^{1/\beta} \leq C \;(\int_{M}...
2
votes
0answers
57 views

square root of a Riemannian metric

Let $(M,g)$ be a compact Riemannian manifold. Taking a smooth vector field $X$, there is an associated smooth function $$g(X,X):M\longrightarrow \mathbb{R}^+, \quad p\longmapsto g_p(X_p,X_p)\geq 0.$$ ...
0
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0answers
49 views

Proof of classification theorem for compact surfaces

I am reading Massey's 'A basic coruse in Algebraic Topology'. In first chapter, he proved classification theorem for compact surfaces (compact connected 2-manifold). This theorem classifies compact ...
3
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0answers
23 views

Can one characterize compact smooth manifolds dynamically as such?

If $M$ is a smooth (connected) compact manifold, then it is known that any (smooth) vector field is complete, which means that the flow exists for all $t \in \mathbb{R}$. What about the converse? If $...
2
votes
1answer
49 views

The relationship between the tubular neighbourhoods of two diffeomorphic manifolds

I'm a beginner of this complex area and want to use the differential geometry as a tool to solve some control problems. So my statement might be a little bit inaccurate...I will try my best. There ...
1
vote
1answer
32 views

Is a pentagon a surface?

My question arise since a pentagon is homeomorphic to a closed disc. This last one is a surface with boundary. However, a pentagon has vertices, so it seems isn't a 2-manifold. If you consider a ...
1
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1answer
57 views

Collar neighborhoods of a topological manifold with boundary

For a $n$-manifold $M$ with nonempty boundary $\partial M$, a collar neighborhood of $\partial M$ in $M$ is an open neighborhood of $M$ homeomorphic to $\partial M \times [0,1)$ by a homeomorphism ...
2
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0answers
41 views

Does every closed manifold have a finite CW structure? [duplicate]

Hatcher's Algebraic Topology Corollary $3.37$ states that a closed manifold of odd dimension has Euler characteristic zero. But to consider about the Euler characteristic, every closed manifold must ...
0
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0answers
21 views

An example of a non-invariant measure on a compact Lie group

I would like to construct a non-invariant measure on a compact Lie group but I'm not sure what is allowed and what the consequences are. Take the simplest example of $SO(2)$. The unnormalized ...
1
vote
1answer
49 views

In what sense are these two invariant measures on $SU(2)$ proportional?

An element $g$ of $SU(2)$ is of the following form: $$ g=\begin{bmatrix} z_1 & z_2\\ -\bar{z}_2 & \bar{z}_1 \end{bmatrix}, $$ where $z_i$ are complex satisfying $|z_1|^2+|z_2|^2=1$. I can ...
0
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0answers
32 views

partition of unity subordinate to an open cover and bump functions

So i was just reading about bump function and i noticed, they always seem to become relevant whenever we consider partitions of unity subordinate to an open cover. The definition of the partition of ...
1
vote
1answer
49 views

Cobordism of points

On the wiki page about cobordism, it is stated that the cobordism of oriented 0-dimensional manifolds is $\mathbb Z$. That seem surprising since One can always draw a line between two points. I ...
3
votes
1answer
73 views

Integral in manifolds problem

Let $M$ be a orientable $n$ dimensional manifold. I'm trying to solve the following assertions: Given a connected system of coordinates $(U,x_1,\cdots, x_n)$, prove that there exists a $n$-form $\...
1
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2answers
41 views

Triangulated manifold implies properties of simplicial complex?

I've been told that a compact $d$-dimensional manifold can be realized as a finite $d$-dimensional simplicial complex. Since any manifold is locally compact (I think), if it can be realized as a $d$-...
1
vote
1answer
146 views

Smooth map with no critical point

We know that given a manifold $M$ that is connected and compact, there exist a real function with a finite number of critical points, and with at least two. Now if we consider a point $x\in M$, is it ...
1
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1answer
58 views

Can someone find the error in my proof that if X is compact then it's a manifold?

So, the question is let $X$ be a Hausdorff space such that each point of X has a neighborhood that is homeomorphic with an open subset of $\mathbb{R}^{m}$. Show that if $X$ is compact, then $X$ is an $...
1
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0answers
37 views

About a sub-manifold of $S^3$ whose boundary only consists of tori

I am reading a paper called "JSJ-decomposition of knot and link complements in $S^3$", written by Ryan Budney. My question does not concern the essence of the paper but a technical fact about 3-...
3
votes
2answers
108 views

smooth functions on compact Lie groups

Let $G$ be a compact matrix (Lie) group. If $$f:G\longrightarrow \mathbb{C}$$ is a smooth function I would like to know if there are finite number of smooth functions $$f_{k}, \hat{f}_{k}:G\...
1
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0answers
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 ...
0
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0answers
44 views

Finding the entropy

Let $M$ be a compact manifold and $F:M\to \mathbb{R}$ Morse function. Let $\phi_t$ be a flow generated by $F$ in the following way: $\frac{d\phi _t}{dt}=-\nabla F(\phi _t)$. Let $f=\phi _1$. Find $h(f)...
1
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2answers
55 views

Compact Connected Manifolds of dimension $n$

I started studying topology and seeing some differential geometry and i found it interesting when the book i was reading said that the only one-dimensional compact connected manifold is the circle, ...
11
votes
1answer
222 views

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 \...
2
votes
1answer
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, ...
2
votes
1answer
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 ...
1
vote
1answer
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 $\mathbb{R}^3$?
8
votes
1answer
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) ...
1
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0answers
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 ...
0
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0answers
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 ...
1
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0answers
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 ...
1
vote
1answer
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 ...
1
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0answers
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 $H^p_k(M)$,...
1
vote
1answer
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 ...
0
votes
0answers
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 ...
2
votes
0answers
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(\...
2
votes
0answers
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}=\...
1
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0answers
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?
1
vote
1answer
91 views

Stiefel Whitney class and intersection form

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 ...
4
votes
2answers
139 views

Why is this 3-manifold irreducible?

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 $\...

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