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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Is this statement about manifold true?

Suppose $M$ is a closed $k-$manifold in $\mathbb R^n$ without boundary, can we always find a smooth function $f:\mathbb R^n\to\mathbb R^{n-k}$ such that $M$ is the level set where $f=0$?
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is there a diffeomorphism with only finite orbits but of infinite order?

Is it possible for a diffeomorphism $\phi$ (of a smooth manifold $M$) to have the following property: All its orbits are finite but $\phi$ is not of finite order? (and hence has arbitrarily large ...
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2answers
164 views

How to Show Cotangent Bundles Are Not Compact Manifolds?

Hamiltonian mechanics occurs in a sympletic manifold called phase space. Lagrangian mechanics take place in the tangent bundle of the configuration manifold. Using Legendre transform makes possible ...
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1answer
51 views

Working with homomorphisms and de Rham cohomology.

Here’s my question: Let M be a connected, compact, orientable, smooth n-manifold ($ n \in \mathbb{N}_{\geq 2} $). Let V be a neighborhood of p diffeomorphic to $\mathbb{R}^n$ and let U = M \ {p}. ...
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73 views

Finding the de Rham cohomology of an open subset of $ \Bbb{R}^{n} $ minus a point.

Here’s my question: Let $ n \in \mathbb{N}_{\geq 2} $. Suppose that $ U \subseteq \Bbb{R}^{n} $ is an open set and that $ x \in U $. Then show that $$ {H_{\text{dR}}^{n - 1}}(U \setminus \{ x ...
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24 views

Tangent space chart dependent or not?

I was wondering whether the definition of a tangent space is chart dependent or not? Cause to me it seems that they depend on the charts: Let $p \in M$ where $M$ is a k-dim smooth manifold and $\phi: ...
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32 views

Is the $C^0$-fine topology finer than the metric topology?

Let $C(E,F)$ be the set of continouos maps between metric spaces $E$ and $F$. Suppose we are given the $C^0$ fine topology and a metric topology on $C(E,F)$. We know that the fine topology is finer ...
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1answer
28 views

how to find points where a k-form is nonvanishing.

for example, if we are given 2-form $\omega=2xdx\wedge dy+2ydy\wedge dz$, what are the points where the form vanishes? I can only think of points $(0,0,z)$, is it all? Additionally, if we have a form ...
4
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1answer
40 views

Can $\mathbb{R}\mathbb{P}^2$ be embedded into an orientable 3-manifold?

We know that $\mathbb{R}\mathbb{P}^2$ cannot be embedded into $\mathbb{R}^3$, but is there an orientable 3-manifold where it is possible?
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1answer
23 views

Extending a diffeomorphism outside a compact set

I believe that the following statement is true: Let $U,V\subset \mathbb{R}^n$ be open sets, $K\subset U$ compact, and $\gamma:U\to V$ a diffeomorphism. Then there is a diffeomorphism ...
3
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2answers
113 views

Is the $G$-action on a principal $G$-bundle proper?

Let $G$ be a Lie group. If $G$ acts properly and freely on a manifold $P$, then it is well-known that $P \to P/G$ form a principal $G$-bundle. I would like to know the converse: namely Question: if ...
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20 views

Show that every k-dimensional vector subspace V of $R^N$ is a manifold diffeomorphic to $R^k$.

I'm actually in a exercise of the book " Differential Topology " of Guillemin and Pollack. Show that every k-dimensional vector subspace $V$ of $R^N$ is a manifold diffeomorphic to $R^k$, and that ...
4
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1answer
175 views

Intuitive meaning of immersions

I have a hard time understanding the concept of immersions. In my course, it was only introduced by the immersion theorem wich says: Let $f: U \subset \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ be ...
4
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1answer
134 views

4-manifold: $0$-handle $\cup$ $2$-handles along a framed link in $S^3$ (intersection form = linking matrix = presentation matrix of $H_1(\partial M)$)

Let $L$ be a framed link in $S^3$, consisting of framed knots $L_1,\ldots,L_m$. Let $A=[a_{ij}]\in\mathbb{Z}^{m\times m}$ be its linking matrix, with $a_{ii}=$ framing of $L_i$ and $a_{ij}=$ linking ...
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33 views

Index of a zero of a normal vector field

Let $M$ be a manifold and $S\subset M$ a submanifold. Assume whatever regularity you need (smoothness, compactenss, orientability, empty boundary, proper embedding of $S$,... this question is ...
3
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1answer
49 views

Difference between Euler characteristics of a Riemann surfaces

Let $X$ be a compact connected Riemann surface of genus $g$. Let $U$ be the complement of $r$ points in $X$. The Euler characteristic of $X$ = $2-2g$. That I understand. But I'm confused about the ...
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1answer
56 views

projective space and torus

we defined the projective space as $\mathbb{S}^2$ with opposie side identification and the torus as $\mathbb{R}^2 / \mathbb{Z}^2.$ And now I am concerned with their manifold structure- In fact, I ...
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23 views

Immersion of punctured torus into Euclidean [duplicate]

(a) Show there is an immersion of the punctured torus $S^1\times S^1$ - {a point} into $R^2$. (b) generalized it to $T^n$ - {a point} into $R^n$ can you give concrete proof for these problem? ...
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1answer
26 views

For a positive definite quadratic form $f: R^n \rightarrow R$, $f^{-1}(x)$, for any $x>0$, is diffeomorphic to $S^{n-1 }$

How to show for a positive definite quadratic form $f: R^n \rightarrow R$, there exists $f^{-1}(x)$, for any $x>0$, is diffeomorphic to $S^{n-1 }$?
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1answer
45 views

What intuition do we have for a subalgebra of Lie to be abelian?

The motivation for my question comes from the definition of rank of a given globally symmetric space: it is based on the image of a maximal abelian subalgebra of a given algebra by the exponential ...
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1answer
25 views

Unsolved regular value problem in $\mathbb{R}^n$

I want to show that if $F : \mathbb{R}^n \rightarrow \mathbb{R}^{n-k}$ is a $C^1$ function and $rank(DF) = n-k$ then $M:=F^{-1}(\{0\})$ defines a manifold. My idea: Without loss of generality I ...
3
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1answer
61 views

Open Unit Ball diffeomorphic to the Open Unit Cube

How can I show that the open unit cube $(-1,1)^n \subset \mathbb{R}^n$ and the open unit ball $B = \{x \in \mathbb{R}^n \mid \|x\| < 1\}$ are diffeomorphic? I know that one can proof this by ...
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1answer
37 views

A differentiable manifold of class $\mathcal{C}^{r}$ tangent to $E^{\pm}$ and representable as graph

The stable manifold theorem tell us: A local stable manifold $W^{s}_{loc}(x^{*})$, which is a differentiable manifold of class $\mathcal{C}^{r}$ and dimension $n_{-}, $ tangent to the ...
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53 views

Embed $S^{p} \times S^q$ in $S^d$?

Can we embed $S^{p} \times S^q$ in $S^d$ with all the nice properties, what are the allowed values of $p$ and $q$ for $d=2,3,4$ where $p+q \leq d$? =For $d=2$= I suppose that we cannot embed $S^{1} ...
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2answers
171 views

Why would one care about Fibre Bundles

As a physics student I can easily understand the motivation for studying manifolds and why the definition looks the way it does, I only have to think of Minkowski space in GR. But for the life of me ...
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0answers
48 views

Momentum a cotangent vector?

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
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22 views

Induced bundles and Smooth Maps

Given a smooth map between compact manifolds without boundary what criteria guarantee that the induced bundle is isomorphic to the tangent bundle? And less generally, suppose the map has non-zero ...
2
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1answer
41 views

immersion of punctured torus in plane

Let $S^1 \times S^1$ be the $2$-torus. If a point $a=(p,q)$ of the torus is removed, i.e., it is punctured at one point then how can I show that it can be immersed in the plane, i.e., in $\Bbb{R}^2$? ...
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88 views

Finite set on compact manifolds

I feel blocked with this claim - it sounds intuitively true, just thinking as a jellyfish entering a real line, the intersection of her legs with the real line is certainly finite since the jellyfish ...
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3answers
63 views

The biggest degree of a map between fixed surfaces

Let $S_1$ and $S_2$ be compact surfaces (real manifolds of dimension 2). None of them is a sphere. Question: What is the biggest degree of a smooth map from $S_1$ to $S_2$? Comment 1. I have a ...
2
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1answer
107 views

Transversality of Vector Fields Defined in terms of Diff. Forms and Open Books.

All: Sorry for the length of the post, but I think it is necessary to set things up so that the post is understandable. I'm trying to understand how it is that the transversality (in this case , the ...
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0answers
37 views

How to prove that the tangent map $T\phi$ into the pullback bundle is smooth?

Assume $\phi: M\rightarrow N$ is smooth. Let $\phi^*(TN)$ be the pullback bundle of $TN$ by $\phi$. Define $T\phi:TM\rightarrow \phi^*(TN)$ as follows: $T\phi(m,v)=(m,d\phi_m(v)) $. We also have the ...
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28 views

On critical values of a linear function $g:SO_2\times \mathbb R^2\to \mathbb R^2$

Consider map $g:SO_2\times \mathbb R^2\to \mathbb R^2$, $g(A,v)=Av$, where $A\in SO_2$ is an orthogonal $2\times 2$ matrix and $v\in \mathbb R^2$ is a $2$-vector. Show that $0$ is a critical value. ...
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1answer
170 views

Different definitions of differential forms?

I am a physicist and was reading about differential forms in Classical Mechanics. Now, I thought that a two-form is a smooth map $\omega : M \rightarrow \Lambda(T^*M)$ so that a point $p$ on the ...
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1answer
97 views

Differential forms - looking for 3 definitions!

I am sorry for this type of question, but I currently have to deal with differential forms although I have not heard so far what they actually are, so I have just a few very particular questions about ...
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2answers
77 views

What are monomorphisms in the category of real vector bundles over a fixed base space $X$?

I have a (perhaps stupid) question dealing with vector bundles. In most textbooks, a morphism of real vector bundles $f:\xi\to \eta$ over a fixed base space $X$ is said to be a ...
2
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1answer
48 views

Diffeomorphism between a sphere and ellipsoid in $\mathbb R^3$.

Diffeomorphism between a sphere and ellipsoid in $\mathbb R^3$. I've been looking for an diffeomorphism between a sphere in $\mathbb R^3$ and an ellipsoid of the form $$\{ (x,y,z) \in \mathbb R^3 ...
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1answer
13 views

How do I see that a homeomorphism $\sigma$ is an open function?

How do I see that a homeomorphism is an open function ? Given a homomorphism $\sigma: X \rightarrow Y$ between topological spaces, how do i then see that $\sigma(V)$ is open in $Y$ for $V$ open in ...
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1answer
39 views

Let $S$ be a regular surface and let $F$ be a diffeomorphism. How can I prove that the image $F(S)$ is a regular surface?

Let $S$ be a regular surface and let $F$ be a diffeomorphism. How can I prove that the image $F(S)$ is a regular surface ? Consider a regular surface $S \subset \mathbb R^3$ and a diffeomorphism ...
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1answer
107 views

How equivalent are the homeomorphism and diffeomorphism groups of 3-manifolds?

Every topological 3-manifold carries a smooth structure, unique up to diffeomorphism. In addition, "up to homotopy", the maps between them are the same: every continuous map is homotopic to a smooth ...
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Teaching myself differential topology and differential geometry

I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology. I have decided to fix this lacuna once for ...
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3answers
532 views

Is Stokes' Theorem natural in the sense of category theory?

Stokes' Theorem asserts that for a compactly-supported differential form $\omega$ of degree $n-1$ on a smooth oriented $n$-dimensional manifold $M$ we have the marvellous equation $$\int_M d\omega = ...
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1answer
39 views

Conifolds and Exotic Spheres

First of all, a disclaimer: I'm a physicist trying to understand mathematical aspects of some solutions I've encountered while studying string theory, and am certainly not a mathematician of any sort. ...
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1answer
36 views

What does “orthogonal complement” mean in Milnor's Topology from the Differentiable Viewpoint?

Milnor writes on p. 11 If $M'$ is a manifold which is contained in $M$, it has already been noted that $TM'_x$ is a subspace of $TM_x$ for $x \in M'$. The orthogonal complement of $TM'_x$ in ...
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1answer
98 views

Torus/Möbius Band homeomorphism

Is a fattened Möbius Spiral Band homeomorphic to a Torus? (Due to the same Euler Characteristic $\chi$ ?) Are both non-orientable? Following (3D printable plastic) Torus has a square section that ...
2
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1answer
115 views

Trivialisation of Möbius strip

I've just started studying Advanced Geometry and I'm in trouble with a (stupid) exercise. It's about finding a trivialisation of the Möbius strip (I'll refer to it as $E $) viewed as a fibre bundle ...
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1answer
30 views

Interior of image of regular points is dense?

I'm studying some problems related to differential topology and I came across the following exercise: if $f:M\rightarrow N$ is a surjective smooth (i.e., $C^\infty)$ function, $\dim(M)>\dim(N)$ and ...
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54 views

a question of my real analysis class,can someone help me solve this question?

Let $n \geq 1$ be an integer, and $C: \mathbb{N}^n \to \mathbb{R}$ be a function. Prove that there exits a infinitely differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ whose value and partial ...
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1answer
41 views

A question about the index of vector field

$M$ is the boundary of a compact manifold $U$: $M = \partial{U}$, $\mathbf{v}$ is a unit vector field on $M$, how to prove that if $\mathbf{v}$ can be extended to be a nonvanishing vector field on all ...
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2answers
50 views

The bundle vector $f^\ast(\xi)$ for Moebius over $S^1$

Take the Moebius band like a vector bundle $\xi$ over the circle $S^1$ and the functions $f_n(z)=z^n$ then my question is: how describe the vector bundle define for the pullback $f^\ast(\xi)$ for ...