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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Grassmanians and boudaries of manifolds

Let $M$ be a smooth, compact manifold without boundary. I will say that $M$ is a boundary when there is a smooth, compact manifold with boundary $W$ such that $\partial W=M$. After some lectures I ...
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1answer
30 views

Understanding Result on Non-Degenerate Critical Points

I read a result in a collected works of Steven Smale and one result leapt out at me which I'm clearly not understanding. Stated: Theorem 1.1 (a): Suppose $J: M \to \mathbb{R}$ is a $C^2$ ...
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A differentiable fiber bundle exists for any family of clutching functions.

Lemma: Let $F$ be a smooth manifold and let $\{U_a\}_{a\in A}$ be a covering of a manifold $B$, and let $\{g_{ab}\}_{a,b\in A}$ be a family of clutching functions. That is, $g_{ab}:U_a\cap U_b\to ...
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1answer
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Is a mapping a homeomorphism

I'm considering the mapping $\Psi: C^2([0,1])$ to $C^1([0,1])$ via: $f(x) \mapsto f(x)+x\cdot f'(x)$. Is this mapping a homeomorphism? It should be continuous given that, for any sequence $(f_n) \in ...
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solubility of nonlinear overdetermined PDE for holonomic scaling of a frame

This question is essentially a tweak of under what conditions can orthogonal vector fields make curvilinear coordinate system? . Suppose we have a frame of vector fields $\nu_i$ on $\mathbb{R}^n$. As ...
2
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The inverse of a smooth bundle map is smooth.

Consider smooth fiber bundles $P_i:E_i\to B_i,\quad i=1,2\quad$ with fiber $F$. Let $\tilde f:E_1\to E_2$ be a smooth bundle map. That is a smooth map which preserves the fiber $F$. Let $f:B_1\to B_2$ ...
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43 views

Tangent space change of bases

Let $M\subset \Bbb{R}^m$ be a $k$-dimensional differentiable submanifold. Let $(\varphi, U)$ and $(\psi, V)$ be two charts for $p\in M$ with $\varphi(x)=p$ and $\psi(y)=p$. Then we have two bases for ...
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19 views

Subset of a smooth manifold

I am actually in the resolution of the problem Show that $\Delta$ is diffeomorphic to X, so $\Delta$ is a manifold if $X$ is. - "Differential topology" of Guillemin and Pollack (my own ...
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32 views

Show that $\Delta$ is diffeomorphic to X, so $\Delta$ is a manifold if $X$ is. - “Differential topology” of Guillemin and Pollack

I know that we can refered to the question How to show that the diagonal of $X\times X$ is diffeomorphic to $X$?. I have the same question with an answer, and I needed that someone tell me if it is ...
2
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25 views

Proof of Reeb's theorem without using Morse Lemma

I'm trying to prove Reeb's theorem as stated in Milnor's Morse Theory. That is, suppose we have an $n$-manifold $M$ together with a smooth function $f$ with exactly two critical points (both ...
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20 views

Show that the projection map $X \times Y \rightarrow X $, carrying $(x,y) \rightarrow x$, is smooth.

How do I show that the projection map from $X \times Y \rightarrow X $, carrying $(x,y) \rightarrow x$, is smooth? A map $f:X \rightarrow R^m$ defined on an arbitrary subset $X \subset R^n$ is called ...
2
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1answer
73 views

Prove or disprove any continous map $f$ from $T^2$ to $RP^2$ is null-homotopic.

I tried to solve the following question: Prove or disprove any continous map f from $T^2$ to $RP^2$ is null-homotopic. We know the universal cover of $RP^2$ is $S^2.$ I want to construct a map $g$ ...
105
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5answers
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Why can't differentiability be generalized as nicely as continuity?

The question: Can we define differentiable functions between (some class of) sets, "without $\Bbb R$"* so that it Reduces to the traditional definition when desired? Has the same use in at least ...
2
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33 views

What can I say of an $m$-dimensional submanifold $S$ of an $m$-dimensional manifold $M$?

I consider a differentiable manifold $M$ of dimension $m$. Let be $S$ a submanifold of $M$ of the same dimension $m$. What can I say about $S$? I have tried to prove that $S$ is open but I get ...
2
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1answer
62 views

Diffeomorphism between $T$ (torus) and the cover $S^1 \times S^1$ - Question of the book '' Differential Topology '' of Guillemin and Pollack

Here's the question : ''The'' torus is the set of points in $\mathbb{R^3}$ at distance $b$ from the circle of radius a in the xy plane, where $0<b<a$. Prove that these tori are all diffeomorphic ...
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34 views

Closed geodesic minimizing properties

Considering closed geodesics on a compact manifold M of even dimension, what does it mean to say that a curve (any closed geodesic) is locally energy minimizing but not globally ? For simplicity, say ...
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35 views

Why gradient-like dynamical systems are special case of Morse-Smale systems?

I'm studying Morse Theory and my question is exactly as stated in the above title. I can't see how a gradient-like dynamical system could be considered as a Morse-Smale system? Thanks in advance for ...
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40 views

The inverse image of any regular value is a submanifold

Let $f:M^m\subset\mathbb R^k\to N^n$ be a smooth map between manifolds with $m>n$. Let $y\in N$ be a regular value for $f$. Let $x\in f^{-1}(y)$ and $L:\mathbb R^k\to\mathbb R^{m-n}$ be a linear ...
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Applications of $C^\ast$ algebras in differential topology

I was wondering if there were any useful ways $C^\ast$ algebras come into play within differential topology. I know this is a fairly broad question,so any type of input is applicable.
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3answers
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Union of the two coordinate axes in $\mathbb{R^2}$ - From the book '' Differential Topology '' of Guillemin and Pollack

I have tried to solve the problem : Prove that the union of the two coordinate axes in $\mathbb{R^2}$ is not a manifold. Let $X = \{(x,y) \in \mathbb{R^2} : x=0~or~ y=0\}$ be the union of the two ...
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1answer
55 views

Topological question from the book '' Differential Topology '' of Guillemin and Pollack

Here's the question : A smooth bijective map of manifolds need not be a diffeomorphism. In fact, show that $$f:\mathbb{R^1}\rightarrow {R^1}$$ $$x\rightarrow f(x)=x^3,$$ is an example. I would like ...
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On the following of the question: every $k$-dimensional vector subspace $V$ of … [duplicate]

This is the continuity of this question I've created this question recently, but I didn't receive all the answers I hoped. Someone could explain me why is it that the problem of approach work? In the ...
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1answer
51 views

Area form a differential form?

I just read that $\omega_x( \eta, \zeta) := \langle x, \eta \times \zeta \rangle $ is the area form on the sphere, where $x \in \mathbb{S}^2$ and $\eta,\zeta \in T_xM.$ All I see is that this is ...
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36 views

Focal points of the parabola $y = x^2$ in $\mathbb{R}^2$. [closed]

Let $X$ be an $n-1$ dimensional submanifold of $\mathbb{R}^n$, a "hypersurface." A point in $\mathbb{R}^n$ is called a focal point of $X$ if it is a critical value of the normal bundle map $h: N(X) ...
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29 views

Immersion except at the origin. [closed]

Whitney showed that for maps of two-manifolds into $\mathbb{R}^3$, a typical cross cap looks like the map $(x, y) \to (x, xy, y^2)$. Prove that this is an immersion except at the origin.
4
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1answer
65 views

Is this statement about manifold true? [duplicate]

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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1answer
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is there a diffeomorphism with only finite orbits but of infinite order?

Note: after not receiving any answer for some time, I asked this in mathoverflow, and got an answer there. The Question: Is it possible for a diffeomorphism $\phi$ (of a smooth manifold $M$) to have ...
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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
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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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0answers
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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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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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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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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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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 ...
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1answer
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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 ...
2
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1answer
58 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 ...
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39 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
57 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 ...
3
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1answer
62 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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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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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 $S^{n-1 }$?
3
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1answer
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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
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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
75 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 ...
2
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1answer
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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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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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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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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
53 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 $\Bbb{R}^2$? ...
4
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1answer
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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 ...