# 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 ...
36 views

### 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....
1 vote
106 views

### 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 $\Theta$. ...
79 views

### $S^1\times S^2$ 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 ...
31 views

### 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 ...
1 vote
19 views

### 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 ...
123 views

54 views

### 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 ...
79 views

### 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. ...
452 views

### 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 ...
1 vote
37 views

### 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? ...
73 views

### 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 ...
102 views

### 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 ...
108 views

### 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 ...
184 views

### 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. ...
43 views

### 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 ...
62 views

### 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 ...
1 vote