Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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Map between Tangent Manifolds Well-Defined?

Let $f: \mathcal{M} \to \mathcal{N}$ be a $\mathscr{C}^{r+1}$ map. We define a map $\mathscr{T}f: \mathscr{T}\mathcal{M} \to \mathscr{T}\mathcal{N}$ as follows: A local representation of the map $\...
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108 views

Why does the slope of a smooth simple closed curve have winding number one?

$\def\RR{\mathbb{R}}$Let $S^1$ be the circle and let $\gamma : S^1 \to \RR^2$ be a smooth injective map with $\gamma'(t)$ everywhere nonzero. What is the easiest way to show that $t \mapsto \gamma'(t)$...
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82 views

Stokes or homotopy?

The problem states Show that if $X$ is a simply connected manifold, then $\oint_{\gamma}\omega=0$ for all closed 1-forms $\omega$ on X and all closed curves $\gamma$ in $X$. However I have ...
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3answers
336 views

Embeddings are precisely proper injective immersions.

We call a map $f: X \to Y$ between topological spaces proper if $f^{-1}(K)$ is compact for all compact $K \subset Y$. Where can I find a reference that embeddings are precisely proper injective ...
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70 views

Embed curves in the plane

The strongest version of Whitney's embedding theorem says that every smooth real $n$-dimensional manifold $M^n$ (Hausdorff and second-countable) can be embedded in $\mathbb{R}^{2n}$. This should mean ...
2
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1answer
257 views

Laplacian and Hodge Laplacian

I am new to the theory of differential forms, but there is one thing that I don't get at all. Imagine that you are on the sphere $\mathbb{S}^2$, then the Laplacian $- \Delta$ is known to be a ...
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1answer
39 views

a generalization of punctured cylinder

Let $S^1\times \mathbb{R}$ be the infinite cylinder. Pucture it, we have $S^1\times \mathbb{R}-*$. Then $(S^1\times \mathbb{R}-*)\simeq Skeleton^1(S^1\times \mathbb{R})\simeq S^1\vee S^1$. How ...
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39 views

Terminology question : “half smooth, half topological” fibre bundle

First, I know (or I think I know...) the definition of fiber bundle, be it in the smooth or topological category. Here is my situation, which is kind of between the two: I have a smooth manifold $E$, ...
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1answer
81 views

cohomology ring of a quotient space

what is the cohomology ring $$ H^*((S^3\times S^3\setminus \{e\})/(a,b)\sim (ab,b^{-1});\mathbb{Z}_2)? $$ Here the unit $e$ and the product $ab$ is of the Lie group $S^3=Sp(1)$.
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122 views

Why are contact structures studied from a cohomological, rather than homological, perspective?

As far as I know, contact structures are studied from a "cohomology/dual" perspective, meaning mostly from the perspective of contact forms and their respective kernels, instead of from a "homological"...
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0answers
73 views

Convention of a continued fraction presentation of a lens space

I want to clarify the following two conventions on a surgery description of a lens space. Let $p$ and $q$ are relatively prime integers. Express $$ p/q=x_1-\cfrac{1}{x_2-{\cfrac{1}{x_3-\dots\cfrac{1}{...
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38 views

Relation between differentials of perturbations of vector fields

Let $A$, $B$ be smooth submanifolds of a smooth manifold $M$ and $X\in C^\infty(TM)$ a vector field such following its flow $\xi^X$ gives a diffeomorphism between $A$ and $B$. Suppose also that $P\...
3
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1answer
89 views

Volume of a paracompact manifold

It is stated, without proof, in Wald (1984) (General Relativity) that given any connected manifold $M$ (which is by definition paracompact), one may define a volume measure $\mu$ such that $\mu[M]$ is ...
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2answers
232 views

Why any short exact sequence of vector spaces may be seen as a direct sum?

This is actually the first time I have worked with short exact sequences and I have no clue why the following assertion is true: Any short exact sequence of vector spaces $$ 0 \longrightarrow U ...
2
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33 views

Differentiability of a map into $\wedge T^*M$

Let $M$ a differentiable manifold and $p \in M$. Consider $(U;x_1, \dots, x_n)$ a coordinated system around $p$. $\{dx_\phi\}_\phi$ with $\phi \in \cal{P}(\{1, \dots n\})$, $dx_\phi=dx_{i1} \wedge \...
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1answer
47 views

A question about a part of the proof of the existence of the exterior derivate.

Let $M$ a differentiable manifold, $(U;x_1,\ldots,x_n)$ a coordinated system and $\omega$ a differential form which domain intersects $U$. In $U \cap \operatorname{Dom} \omega$, $\omega$ can be ...
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1answer
37 views

Schwartz rule in differentiable manifolds.

Let $M$ a differentiable manifold and $(U,\varphi)$ a chart with coordinate functions $(x_1,...,x_n)$. Let $p \in U$. Given $f:U\longrightarrow \mathbb{R}$, $f \in C^\infty(U)$, it is possible to ...
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97 views

Explicit Calculation of the Euler class for the 2-Sphere using transition functions

I have been trying to learn about characteristic classes for months now, and every time I try a simple example something goes wrong. Any insights would be greatly appreciated. I am trying to follow ...
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0answers
144 views

Integral Curve of the vector field

If we have a 2-sphere with coordinates $x=r \cos \theta \cos \phi$, $y= r \sin \theta \sin \phi$ and $z=r \sin \theta$ and the vector field $X= (-r\sin \theta \cos\phi, r \cos \theta \sin \phi, r \cos ...
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1answer
57 views

Locally exact vs globally exact

Why the volume form in Sphere is locally exact but not globally exact? here the integral is integral $$\int_{S^n}w$$ with $$w = \frac{1}{r} \sum_{i=1}^{n+1} (-1)^{i-1} x_i dx_1 \cdots\hat{dx_i},\cdots ...
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2answers
73 views

Proof a $(2n-1)$-compact manifold

I have no idea how prove that $$\{(z_0,\ldots,z_n)\in\mathbb{C}^{n+1} \quad| \quad z_0^d+z_1^2\ldots+z_n^2=0, \quad |z_0|^2+|z_1|^2\ldots+|z_n|^2=2\}$$ is a $(2n-1)$-compact manifold. How give the ...
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0answers
55 views

How to prove that $T\mathbb{S}^3 \to \mathbb{S}^3$ is a trivial 3-vector bundle? [closed]

How to prove that $T\mathbb{S}^3 \to \mathbb{S}^3$ is a trivial 3-vector bundle?
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0answers
111 views

Making a gradient-like vector field a gradient vector field via choosing a Riemannian metric.

Let $\xi$ be a vector field on manifold $M^n$ which is a gradient-like vector field for a some Morse function $f$. Prove that there exists a Riemannian metric on $M$ such that $\xi$ is a gradient ...
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2answers
47 views

Covering groups of compact connected Lie groups is compact.

Hi: I have a question as follows: Is it true that the covering group of a compact, connected Lie group is also compact? Thanks very much!
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1answer
61 views

last step in proof of existence of coordinate vector field

This is problem 7 on page 172 of Spivak's Differential Geometry pt. 1. Given a smooth manifold $M$ and a smooth vector field $X$ on $M$, Check that if the coordinate system $x$ is $x = \chi^{-1}$ ...
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1answer
46 views

Is the smooth map with constant rank dense?

Let $M$ and $N$ be two Riemannian manifold. Assume that $f:M\rightarrow N$ is a smooth map and $\dim M < \dim N$. Can we find a smooth map $g$ $C^{k}$-close to $f$ which is immersive at each point ...
2
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1answer
99 views

What is a regular homotopy?

The definition of regular homotopy from Wikipedia says that two immersion $f,g:M\to N$ are regularly homotopic if they represent points in the same path-component of $\text{Imm}(M,N)$. What does "...
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1answer
106 views

Is it possible for a manifold to have a normal vector that is zero everywhere, if so, would this indicate that the manifold is non-orientable?

Basically I've been thinking about defining a non-orientable three-dimensional metric space via defining the normal vector and looking to see if there is two possible vectors for the same point. I'm ...
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1answer
108 views

Find the Poincare Dual of a ray in $\mathbb{R}^2-\{0\}$

This is example-exercise 5.16 in Bott and Tu (which I'm independently reading through.) The problem states: Let $M=\mathbb{R}^2-\{0\}$, and $X\subseteq M$ be the closed submanifold $\{(x,0):x>0\}$....
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1answer
91 views

What is the definition of a properly-immersed arc?

I am going over a paper in which a theorem includes a condition that certain arcs are "properly-immersed geodesic arcs" in a surface $\Sigma$ with nonempty boundary. I have done a search, but I have ...
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2answers
109 views

Morse function on $\mathrm{SO}(n)$

I would like to prove that the following function is a Morse function, $F : \mathrm{SO}(n)\to\mathbb{R}$ $$A=(a_{ij})\mapsto\sum_{i=1}^na_{ii}\lambda_i$$ with $0<\lambda_1<...<\lambda_n$. ...
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1answer
111 views

Euler characteristic of the closed unit ball

I would like to calculate the Euler-Poincaré characteristic of the closed unit ball $B$ by the de Rham cohomology, the Poincaré-Hopf theorem and Morse theory. de Rham cohomology: since $B$ is ...
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209 views

Applying the Frobenius theorem to a decomposable 2-form

So I have the following problem: Suppose $\omega=\phi \wedge \theta$ is a closed decomposable 2-form on $M$ a manifold (decomposable just means it can be written as a wedge of 1-forms). Suppose $p\in ...
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1answer
87 views

Proof Transverse Submanifolds

Let $f:M\rightarrow\mathbb{R}^p$ differentiable and $N$ submanifold of $\mathbb{R}^p$. Show that for all $\epsilon>0$ exist $v\in\mathbb{R}^p$ whit $||v||<\epsilon$ such that $f(x)=f(x)+v$ is ...
4
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1answer
102 views

Completeness implies geodesic completeness, a more conceptual way?

We know from Riemannian geometry that for Riemannian manifolds, completeness and geodesic completeness are equivalent, which is usually a consequence of Hopf-Rinow theorem. However, I'm considering a ...
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2answers
92 views

Actual Classification re Nielsen-Thurston Theorem (how to)?

according to Nielsen -Thurston Classification: http://en.wikipedia.org/wiki/Nielsen%E2%80%93Thurston_classification If $S$ is compact and orientable surface, then any homeomorphism is isotopic to (...
3
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1answer
71 views

mutually transverse embedded submanifolds, natural bundle surjections, direct sum, isomorphism

Let $N$ be a manifold and let $M_1, \dots, M_n \hookrightarrow N$ be mutually transverse embedded submanifolds, so $M = \cap M_i$ is an embedded submanifold of $N$ with $\text{T}_m(M) = \cap T_m(M_i)$ ...
2
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1answer
256 views

A non-vanishing one form on a manifold of arbitrary dimension

So the problem I have is: Let $\theta$ be a closed 1-form on a compact Manifold M without boundary. Further suppose that $\theta \neq 0$ at each point of M. Prove that $H^{1}_{dR}(M)\neq 0$. The ...
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1answer
52 views

A diffeomorphism with $\det(df)>0$ and convexity of matrices

Let $f: \mathbb{R}^{n}\rightarrow \mathbb{R}^n$ be a diffeomorphism such that $\det(df)>0$ at the origin (hence everywhere). Show that there is a continuous map $F: [0,1]\times \mathbb{R}^{n} \...
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1answer
249 views

Laplace-de Rham operator

Consider an operator $\partial = (-1)^k \star^{-1}\,d\star: \Omega^k(\mathbb{R}^n) \to \Omega^{k-1}(\mathbb{R}^n)$. Note that we equivalently can write $\partial = (-1)^{nk + n + 1} \star\,d\star$. ...
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1answer
41 views

The Projection $S^{2}\times S^{3} \rightarrow (S^{2}\times S^{3})/(S^{2} \lor S^{3})$

So a problem I have come across involves constructing some smooth degree 1 maps, and one such map to be constructed is a map $S^{2}\times S^{3} \rightarrow S^{5}$. I've been told that there is such a ...
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2answers
201 views

Prove that $TTM$? is a orientable bundle on $TM$

I know that $TM$ is always orientable bundle. But what is $TTM$? How try to prove that $TTM$ is a orientable bundle on $TM$.
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1answer
1k views

Properly discontinuous action: equivalent definitions

Let us define a properly discontinuous action of a group $G$ on a topological space $X$ as an action such that every $x \in X$ has a neighborhood $U$ such that $gU \cap U \neq \emptyset$ implies $g = ...
4
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1answer
84 views

Do unit normal vectors of extremal points on a compact hypersurface have opposite directions?

Problem To be definite, by a hypersurface of $\mathbb R^{n+1}$, I mean a connected submanifold of $\mathbb R^{n+1}$ of dimension $n$ without boundary. Suppose $M$ is a compact smooth orientable ...
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1answer
80 views

Homotopy class of maps to a complex projective space

Let $M$ be a closed oriented smooth 4-manifold. Denote by $[M, \mathbb{C}P^{\infty}]$ homotopy classes of continuous maps from $M$ to $\mathbb{C}P^{\infty}$. I would like to know how to show $$ [M, \...
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1answer
128 views

What is the boundary of tubular neighborhood of a compact manifold?

What is the boundary of tubular neighborhood of a compact manifold? for example what is the boundary of tubular neighborhood of a torus $\mathbb{T}^2$ in $\mathbb{R}^3$ or a torus $\mathbb{T}^2$ in $\...
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1answer
87 views

Two definitions of embedded submanifold

Let $N$ be a smooth manifold. One possible definition (I believe) for an embedded submanifold of $N$ is some $M \subset N$ that is a (smooth) manifold such that the inclusion $i : M \hookrightarrow N$ ...
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1answer
114 views

Reference to finite coverings causing injections on deRham cohomology

So, I've heard that if you have a finite degree covering of a compact connected manifold by another compact connected manifold of dimension $n$ (So $\pi :M \rightarrow N$) gives an injection on the ...
2
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2answers
92 views

When are the exterior derivative and contraction of forms inverses?

I am trying to get a better feel for both the exterior derivative of a form and the contraction of a form by a vector field $X$. Basically, when are these inverses? If I have a one-form $\omega$ and ...
2
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2answers
149 views

Universal Cover of a Surface with Boundary. What does Cantor set on Boundary Correspond to?

I am trying to understand in more detail the answer to: Universal Cover of a Surface (with Boundary) It is mentioned that the universal cover of a hyperbolic surface $S$ with geodesic boundary is a ...