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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Understanding topological and manifold boundaries on the real line

Let $M$ be the subset $[0,1)$ $∪ $ {$2$} of the real line. Find its topological boundary $\mathrm{bd}(M)$ and its manifold boundary $\partial M$. I know that to find the topological boundary, I ...
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Parametrization of $n$-spheres

This comes from Guillemin and Pollack's book Differential Topology. The book claims that one cannot parametrize a unit circle by a single map. I thought we could (by a single angle $\theta$). I ...
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Question about index of critical points.

I don't really understand what index of a critical point is and I am trying to do a very simple example. I was wondering if someone could help me figure out what the index of the critical point ...
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Let $A$, $B$ be subsets of $S^n, n≥2$. Show that if $A$ and $B$ are closed, disjoint, and neither separates $S^n$, then…

Let $A,B$ be subset of $S^n$, $n\geq 2$. Show that if $A$ and $B$ are closed, disjoint, and neither separates $S^n$, then $A\cup B$ does not separate $S^n$. I've thought to do it by contradiction and ...
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Does Differential Topology or Differential Geometry play a larger role in Chaos Theory?

I'm an undergraduate on somewhat of a time constraint in school. I have room in my remaining schedule for a semester of either Differential Geometry or Differential Topology. I understand the ...
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120 views

When is a topological space a manifold?

I'm looking for someone to point me in the direction of papers or books which discuss when a topological space (perhaps with the conditions locally compact, Hausdorff) is a topological manifold, ...
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Relationship between trace of a linear map and the number of points it fixes.

Problem Statement: Let $\Phi_A:T^2\rightarrow T^2$ be a smooth mapping into the torus induced by a linear map $A\in SL_2(\mathbb{Z})$ under the quotient relation that identifies 0 and 1. Assume that A ...
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51 views

Extending curves

I have the following situation, $N$ is a $k$-manifold, $X$ a compact $(k+1)$-manifold and $F:X \to N$ a smooth map. Let $y$ be a regular value of both $F$ and $F|_{\partial X}$, hence $F^{-1}(y)$ is a ...
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How can I align the angle between points with the magnetic heading as the points move?

I have 3 robots which must track a point. The distance between all the robots and the point is known so a triangle can be formed between any 2 robots and the point. If I find the angles in the ...
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Why is $\theta \not \in C^{\infty}(S^1)$?

Why is $\theta \not \in C^{\infty}(S^1)$? I know that since $\int_{S^1} d\theta = 2\pi$ then $d\theta$ is not exact. Thus since $d(\theta)=d\theta$, $\theta$ must not be $C^{\infty}$ but it seems ...
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373 views

An alternative description of the first Stiefel-Whitney class

I've heard this description of the first Stiefel-Whitney class of a vector bundle but I don't know why this is true. Can anyone help me with this, please? The first Stiefel-Whitney class of a vector ...
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69 views

How to make a $C^1$ knot into a $C^\infty$ knot

Suppose I have a $C^1$ imbedding $f: S^1 \rightarrow S^3$. From the point of view of knot theory, what's the "best" way to get a $C^\infty$ curve that "looks like" or is "equivalent to" $f$? For ...
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197 views

“Completing” a vector field on a non-compact manifold $M$

Suppose I have a non-compact smooth manifold $M$ and an arbitrary nowhere-vanishing smooth vector field $X$ on $M$ which is not complete. Is there a way to create a smooth vector field $V$ that is ...
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248 views

Complete non-vanishing vector field

Let $M$ be a non-compact smooth manifold. Suppose we have a nowhere-vanishing smooth vector field X. Is this vector field complete? I know it is when $M$ is compact. However, I am unsure in the ...
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60 views

normal form of an n-form

It is known, that one can convert any function $f(x_1,\dots,x_n)$, defined near $0$, into the function $(y_1,\dots,y_n)\mapsto a+y_1$, by a suitable local change of coordinates, provided $df\neq 0$. ...
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58 views

Prove $X =\left \{(x, y) \in \mathbb{R}^3 \times \mathbb{R}^3 \ | \ |x| = 1, |y| = 1, x\cdot y = \frac{1}{2}\right\}$ is a manifold

I am having trouble with the following qualifying exam problem and I would appreciate any help. Thank you. Let $X$ be the set of pairs of unit vectors $(x, y)$ in $\mathbb{R}^3$ such that $x \cdot y ...
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Orientability of $P_{\bf R}T{\bf RP}^{2n}$

I know the following fact : (1) $ {\bf RP}^{2n}$ is non-orientable. (2) $ {\bf RP}^{2n-1}$ is orientable. (3) $P_{\bf R}T{\bf RP}^{2n}$ is orientable. (4) $P_{\bf R}T{\bf RP}^{2n+1}$ ...
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Question about a specific case of the argument principle for maps of circles.

Problem Statement: Let $f:S^1\rightarrow S^1$ be a smooth map of manifolds where $S^1=\frac{[0,1]}{0~1}$, and let $f'(t)\in \mathbb{R}$ be given by the $df_t[1]_t=f'(t)[1]_{f(t)}$ at each $t\in S^1$. ...
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Extending a smooth map

When can I extend a smooth map $f:\mathbb{R^2}-\lbrace 0 \rbrace \to S^1$ to a smooth map $\tilde{f}:\mathbb{R^2} \to S^1$. For instance, consider $g(x,y)=(x,y)/\sqrt{x^2+y^2}$? Am I able to extend ...
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Given a local diffeomorphism $f: N \to M$ with $M$ orientable, then $N$ is orientable.

Given a local diffeomorphism $f: N \to M$ with $M$ orientable. Why is $N$ orientable? My professor wrote this in class without giving a proof and said "you should try to prove this for fun :)". I ...
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“Product” bundle notation.

Let $\newcommand{\Spin}{\operatorname{Spin}}M$ and $M'$ be two manifolds, equipped with a principal $\Spin_n$ and $\Spin_{n'}$ bundle called $P$ and $P'$, respectively. Then there is an induced ...
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What role does differentiability play in Topology?

My question is stated in the title. As a brief background, I'd like to say I know next to nothing about Topology. The little bit I was exposed to came as an aside in my Multivariate Calculus class; we ...
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59 views

Compactness of covering space

If we have space $X$ with and $n$ sheeted covering space $Y$ is $Y$ compact iff $X$ is? Torus or sphere, make me believe the answer is yes.
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145 views

How to prove that the map is open?

I am trying to prove that $\phi$ is homeomorphism,where $U=\{[1,u,v]|u,v\in \mathbb{R}\}\subset{\mathbb{R}P^{2}}$,and $\phi$:$U\rightarrow\mathbb{R}^{2}$ given by ...
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Given that $X$ is closed and $Y$ is connected, prove that $Y$ is also closed.

I am having trouble with the following qualifying exam problem. Suppose $f: X \rightarrow Y$ is a smooth immersion between smooth manifolds of the same dimension. Given that $X$ is closed and $Y$ ...
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Vector field on an odd sphere

Let $x^1,y^1,\ldots,x^n,y^n$ be the standard coordinates on $\mathbb{R}^{2n}$. The unit sphere $S^{2n-1}$ in $\mathbb{R}^{2n}$ is defined by the equation $\sum_{i=1}^n(x^i)^2+(y^i)^2=1$. Show that ...
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Finding the kernel of Pushforward of $f:\mathbb R^n\rightarrow \mathbb R^k$

Let $U$ be an open subset of $\mathbb R^n$, $f:U\rightarrow\mathbb R^k$ a smooth map such that its pushforward is onto, for each $x\in U$, i.e. $$f_{*x}:T_xU\rightarrow T_{f(x)}\mathbb R^k$$ is ...
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Uniqueness of Smoothed Corners

Let $M^m$ be a smooth manifold with boundary. We may form a new topological manifold by "adjoining a handle to $M$", I.e. by choosing a smooth map $h : \partial \mathbb{D}^{\mu} \times ...
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Let $M$ and $N$ be smooth manifolds and $f: M\rightarrow N$ a diffeomorphism. Prove that the map $df:TM \rightarrow TN$ is a homeomorphism.

I am going through qualifying exam questions and I am stuck on this problem. I don't think it should be too difficult, but I am having a lot of difficulty. I am not even sure how to start. Some ...
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181 views

Qualifying Exam Question on Manifolds

I am practicing qualifying exam problems and I am having trouble with the following question. Any help is greatly appreciated. Let $P$ be a polygon with an even number of sides. Suppose that the ...
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217 views

A surjective map which is not a submersion

Is there an example of a smooth map between smooth manifolds which is surjective, but not a submersion? I feel there can't be one, but don't know of a proof. Nor do I know of a counter-example. ...
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Topological space M with partition of unity--->M paracompact. John Lee Problems

Suppose $M$ is a topological space with the property that for every open cover $X$ of $M$, there exists a partition of unity subordinate to $X$. Show that $M$ is paracompact.
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135 views

Show that 2 sets are not homeomorphic

Prove that a closed interval $A=[0,1]$ and $B=\{(x,y)∈R^2 \mid ||(x,y)||≤1\}$ are not manifold I'm struck with this problem.Can anyone explain how and what property should i use to show that for any ...
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215 views

How to show that open interval is manifold but closed one is not

Prove that we can define manifold's structure for $1.$ An open interval $A=(0,1) $ $2.B=\{(x,y)\in R^2 | ||(x,y)||<1\}$ And that we can't define manifold's structure for $3.$ An closed interval ...
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Finding the degree of a map

I am having trouble computing the degree of a certain map using the fact that $f: N \rightarrow M$ where $M$ and $N$ are both $n$-dimensional manifolds induces a homomorphism between the nth ...
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Find an closed 1-form on $\mathbb{R}^2 \backslash (0,0)$ that is not exact.

I need help with the following problem. I am not sure how start and I would be very appreciative if someone could help me with this (I believe easy?) example. Find an closed 1-form on ...
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Why is $\partial\partial M=\varnothing$?

Why is the border of the border of an oriented differentiable $n$-dimensional Manifold $M$ empty, that is $$\partial\partial M = \emptyset?$$
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de Rham comologies of the $n$-torus

I'm attempting to calculate the de Rham cohomologies of the $n$-torus: $n \choose k$. I'd like to use a Mayer-Vietoris sequence relating $H^kT^{n}$ to $H^kT^{n-1}$ and $H^{k-1}T^{n-1}$ so I can use ...
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Example of a diffeomorphism of class $C^{k}$ which is not $C^{k+1}$

Can anyone give me an example of a map $f:\mathbb{R}\to\mathbb{R}$, which is a diffeomorphism of class $C^{k}$ but it is not a diffeomorphism of class $C^{k+1}$?
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Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both Möbius strips?

This question follows up this previous question, which has an accepted answer that I am having trouble believing. (Edit: the answer has now been fixed; thanks Georges!) I have been working on giving ...
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Existence of Solution: Embedding from 2D Euclidean space to a circle

Given a real matrix $X$ with $n$ rows and 2 columns, can the matrix be transformed to a real matrix $Y$ such that all the points formed by the rows of $Y$ lie on a circle (2d) and their inter-point ...
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An example of a differentiable manifold class $C^k$ but not class $C^{k +1} $

I'm looking for an example of a differentiable manifold of class $C^k$ but not class $C^{k +1}.$ I found an exercise in Hirsh's book, which suggests that the graph of $f (x) = |x|^{\lambda}$, where ...
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Extending metrics

Let $\pi:E\to M$ be a rank $k$ vector bundle over the compact manifold $M$ and let $i:M\hookrightarrow E$ denote the zero-section. Then we have a splitting of the restriction of $TE$ to the ...
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Conditions for a projection of a Knot to be a Knot diagram.

friends. I'm working on a problem, the broad scope of which is to show that given a map $f:S^1\rightarrow \mathbb{R}^3$ be a smooth embedding, and a projection map $\pi_v:S^2\rightarrow P_v$, where ...
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183 views

How to check whether a vector field is Morse-Smale?

Setup and notation: Let $f:M\to \mathbb{R}$ be a Morse-function on the compact $m$-dimensional manifold $M$ and let $X$ be a gradient-like vector field for the function $f$. Denote the unstable ...
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264 views

Tangent space to a product

Can you explain this question explicitly. This is a little bit difficult for me, but I want to learn how to solve. Thank you for help. If $M$ and $N$ are manifolds, let $\pi_1:M\times N\to M$ and ...
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150 views

How does degree theory imply that this mapping $f$ is locally onto?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a smooth vector field ($\mathcal{C}^1$ mapping). Let $0$ be a critical point of $f$, i.e. $H f(0) = 0$. Assume that the index of $f$ at $0$ is ...
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Why the tangent bundle of a smooth manifold is an oriented manifold?

I need help with the following question. I am not sure how to begin. Any help will be appreciated. Thank you! For any smooth manifold $M,$ the tangent bundle $TM$ is an oriented manifold.
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intersection on manifold with boundary and relative long exact sequence in homology

Let $(M,\partial M)$ be a connected compact oriented 3-manifold with torus boundary. Let $j: M \to (M, \partial M)$ and $i: \partial M \to M$ be inclusions. We get a long exact sequence ...
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Discretizing continuous surfaces into semi-regular polygons

I am aware that there have been many works on the problem of discretizing a surface into polygons, however, I wonder if in any work the problem of doing so to get polygons with edges of the same ...