Questions tagged [compact-manifolds]

For questions regarding the structure and properties of compact manifolds.

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Simplicial Homology Groups of Circle Wedge a Torus

Compute the simplicial homology groups of $S^1 \vee (S^1 \times S^1)$ in all dimensions. I'm trying to practice simplicial homology, and want to make sure I understand at a technical level what's ...
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Compact Riemann surface is sequentially compact.

Now, I try to prove that; M:a compact Riemann surface. $\forall \{P_j\}_{j\in N}\subset M$ (sequence of points) $\exists\{P_{j_k}\} _{k\in N}$ (subsequence of $\{P_j\}$) s.t. the subsequence converge....
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Fréchet derivative of the total variation norm for measures on a manifold

Let $\Theta$ be a compact $d$-dimensional Riemannian manifold without boundary and $M(\Theta)$ (resp. $M_+(\Theta)$) denote the set of signed (resp. nonnegative) finite Borel measures on $\Theta$. ...
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$S^1\times S^2$ embedded in $\mathbb{R}^4$ [duplicate]

It's easy to embed $S^1\times S^2$ in $\mathbb{R}^5$, since $S^1\subset \mathbb{R}^2$ and $S^2\subset\mathbb{R}^3$, but $S^1\times S^2$ lives also in $\mathbb{R}^4$. How can we write the embedding map ...
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Classify surface given by $abca^{-1}b^{-1}c^{-1}$

I'ven been solving problems from my Topology course, and don't understand something I saw while reading my solved examples. Here's a problem that will let me show my point: Given $X$ a compact ...
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Proof that a compact surface has a tangent plane orthogonal to position

Given a compact surface $S$, is it true that $S$ has a tangent plane that is orthogonal to the position vector for at least one of its points? I believe that the statement is true, but I'm having ...
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1 answer
123 views

Subharmonic Functions on Closed Manifolds are Constant?

Apologies in advance if this question is trivial! Let M be a closed, connected, oriented, n-dimensional, manifold without boundary. Let $f$ be a smooth function on M where $\Delta f \geq 0$, where $\...
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Regularity of higher order elliptic problem on compact smooth manifolds with boundary

I have trouble in finding a source in the literature for the following result: Let $\overline{M}$ be a compact smooth manifold of dimension $n \in \mathbb{N}$ with interior $M$ and non-empty boundary $...
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How to deal with boundary orientation in Exercise 16-5 from Lee's Introduction to Smooth Manifolds

16-5. Suppose $M$ and $N$ are oriented, compact, connected, smooth manifolds, and $F,G:M\to N$ are homotopic diffeomorphisms. Show that $F$ and $G$ are either both orientation-preserving or both ...
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Extending an embedding with trivial normal bundle

Let $M^{m}$ and $N^{n}$ be $C^{\infty}$ compact manifold with boundary (eventually $\emptyset$) and let $j:M \to N$ be an embedding such that the normal bundle of the embedding $\nu(j)$ is trivial. ...
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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 ...
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Can a connected closed manifold strictly contain a closed manifold of the same dimension?

A connected closed manifold can contain another one as a proper subset: for instance, the $1$-sphere (circle) is contained in the $2$-sphere. Is it possible with manifolds of the same dimension? ...
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Irreducible triangulations of manifolds

Does there exist a closed Riemann manifold $M$, two distinct irreducible triangulations $S_1$ and $S_2$ of $M$, and a triangulation $T$ of $M$ such that there exists a sequence of edge contractions on ...
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Graph embeddings on nonorientable surfaces

If a finite graph $G$ can be embedded on an orientable surface of genus $n$, does this mean that it can be embedded on a nonorientable surface of genus $n$? Is the converse of this statement true?
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Prove that if $f:M\rightarrow\Bbb R$ is a scalar function over a 1-manifold M without boundary then $\int_M df=0$

Well James Munkres in the text Analysis on Manifolds prove the general Stoke's theorem for $k$-form when $k>1$ and then he proves it for $k=1$ only when the bounary of the Manifold is not empty and ...
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Manifolds with Euler characteristic equal to $\pm 1$

A compact connected smooth surface has Euler characteristic equal to $\pm 1$ if and only it is homeomorphic to the real projective plane or the connected sum of $3$ real projective planes. What are ...
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Can dimension of manifolds be understood similarly to dimension of schemes?

I’m only beginning to learn about schemes, but I know that at least in some cases, the dimension of a scheme (or variety) is 1 less than the length of the longest chain of irreducible closed subsets. ...
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Smooth identification of the complement of a disk

I was wondering, to costruct a map from a compact smooth manifold $M$ of dimesion $n$ to the sphere $\mathbb{S}^{n}$ of degree $1$, apparently, the most common idea is to wrap a disk $D$, neighborhood ...
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Intuitive explanation of degree

I do understand the for $f : \mathbb{S}^{1} \longmapsto \mathbb{S}^{1}$ the degree of a map (thinking $f$ as a closed curve $\gamma$ defined on $[0,1]$) can be seen as "how many times a closed ...
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degree of gauss map for genus $g$ torus in $\mathbb{R}^{3}$

I read that the degree of the gauss map for a $M$ compact orientable $2-$manifold (connected to use the fact that those are only the $g-$torus) in $\mathbb{R}^{3}$ should be $(1-g)$, which is $\frac{1}...
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13 votes
5 answers
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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 ...
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3 votes
1 answer
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Compact hyperbolic three-manifold - a question

In a recent paper the authors considered a spacetime described by $$AdS_4 \times \Sigma_3 \times \mathcal{I}_r \times S^2$$ where $\mathcal{I}_r$ is an interval of the $r$-coordinate and the two-...
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3 votes
2 answers
217 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. ...
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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 ...
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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 ...
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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) $...
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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=\...
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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 ...
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2 answers
238 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 ...
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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 ...
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1 answer
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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 $M$ ...
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4 votes
0 answers
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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 ...
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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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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}...
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2 votes
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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 $$g(X,X):M\longrightarrow \mathbb{R}^+, \quad p\longmapsto g_p(X_p,X_p)\geq 0.$$ ...
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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 ...
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3 votes
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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 $...
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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 ...
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1 vote
1 answer
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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 ...
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1 vote
1 answer
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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 ...
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2 votes
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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 ...
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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 ...
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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 ...
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1 answer
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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 ...
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3 votes
1 answer
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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 $\...
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1 vote
2 answers
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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 $d$-...
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1 vote
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310 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 ...
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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 $...
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1 vote
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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-...
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3 votes
2 answers
182 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\...
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