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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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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1answer
129 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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1answer
206 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 ...
2
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2answers
169 views

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 ...
3
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1answer
105 views

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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184 views

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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1answer
436 views

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 ...
7
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2answers
164 views

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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184 views

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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76 views

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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0answers
129 views

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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99 views

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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117 views

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 ...
5
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1answer
177 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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1answer
256 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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2answers
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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1answer
95 views

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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117 views

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 ...
2
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2answers
102 views

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 ...
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1answer
101 views

Approach topological manifolds with smooth manifolds

Because I'm doing some problems that consider all the manifolds while the situation is really clear when considering only smooth manifolds. Thus my question is can we always appoint a topological ...
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1answer
52 views

Suspension of $\mathbb{R}P^2$ Contractible?

Is the suspension of $\mathbb{R}P^2$ Contractible? And if it is, How would you prove it. Thank you!
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1answer
94 views

Differential Topology Question on Complex Projective Space

This question seems like it would be very hard to do directly. I wouldn't know where to begin. I was wondering if anyone had a very slick proof of this. The only thing I think is easy is that its ...
2
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1answer
33 views

Matrix Manifolds Question

I am not sure at all how to do the following question. Any help is appreciated. Thank you. Consider $SL_n \mathbb{R}$ as a group and as a topological space with the topology induced from $R^{n^2}$. ...
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53 views

Question Concerning the classification of 1-manifolds

I am having trouble proving the following statement used in proving the classification of 1-manifolds. Any help would be great. Thank you. Let $L$ be a subset of $X$ diffeomorphic to an open ...
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201 views

A covering map from a differentiable manifold

Let $p: C \to X$ is a covering map. Suppose that $C$ is a differentiable manifold. Is X - differentiable manifold? More precisely, I am interested in the case where $C$ is Submanifold of Lie algebra, ...
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1answer
241 views

Lie Groups induce Lie Algebra homomorphisms

I am having a difficult time showing that if $\phi: G \rightarrow H$ is a Lie group homomorphism, then $d\phi: \mathfrak{g} \rightarrow \mathfrak{h}$ satisfies the property that for any $X, Y \in ...
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2answers
135 views

Showing that the exponential map $\mathrm{exp}:\mathfrak{sl}(2,\mathbb{R})\to\mathrm{SL}(2,\mathbb{R})$ is not surjective

I am having a difficult time showing that the exponential map $\mathrm{exp}: \mathfrak{sl}(2, \mathbb{R}) \rightarrow \mathrm{SL}(2, \mathbb{R})$ is not surjective. I have, however, worked out that ...
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1answer
249 views

Exponential map and the special orthogonal group

I need to show that the map exp$: \mathfrak{so}(2) \rightarrow SO(2)$ is surjective. I already have that $\mathfrak{so}(2) = \{A \in M(2, \mathbb{R}) \ | \ A^T + A = 0\}$ and the map is given by ...
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1answer
57 views

Vector Fields Question 4

I am struggling with the following question: Prove that any left invariant vector field on a Lie group is complete. Any help would be great!
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1answer
123 views

Another question on Orientation Preserving Maps

I am stuck on the following question. Sorry about the bad latex skills. Not sure what went wrong. Is the map $f:S^n \rightarrow S^n$ orientation preserving? I constructed an atlas on $S^n$ ...
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1answer
210 views

Is manifold mapping degree equal to algebraic degree for polynomials?

If $M$ and $N$ are oriented $n$-manifolds and $f: M \to N$ then the degree of $f$ is given by $$ \deg f = \sum_{p \in f^{-1}(q)} sign_p f $$ where $q$ is a regular value and the sign is $+1$ if $f$ is ...
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1answer
93 views

Question about Lie Groups

I am having trouble with the following Lie Algebra question. I will appreciate any help greatly. Any Lie group homomorphism $\phi : G \rightarrow H$ is determined by the induced Lie algebra ...
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0answers
114 views

Relations between elliptic curves and topological quantum field theory

I heard that there are relations between elliptic curves and topological quantum field theory (TQFT). I googled and found that something called "elliptic genus" might be the key word to relate these ...
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72 views

Smoothing corners of a handle attachment

Say we attach a $\lambda$-handle, $\mathbb{D}^\lambda \times\mathbb{D}^{\mu}$, to a smooth manifold $M, \partial M$ by simply taking the quotient $M \cup_h \mathbb{D}^\lambda \times\mathbb{D}^{\mu}$ ...
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1answer
43 views

Topological degree of a map with finite energy

Suppose that $\phi:\mathbb{R}^3 \to S^2$ is of class $\mathscr{C}^1(\mathbb{R}^3\setminus \left\{a\right\}) \cap \mathscr{C}^0(\mathbb{R}^3\setminus \left\{a\right\})$, that is $\phi$ might have a ...
4
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1answer
129 views

On the smooth structure of $\mathbb{R}P^n$ in Milnor's book on characteristic classes.

This question concerns problem 1-B in the book of Milnor and Stasheff part a. They first define the set $F:=\{f:\mathbb{R}P^n\rightarrow\mathbb{R} \mid \text{$f\circ q$ is smooth}\}$ where ...
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64 views

smoothness structure on a set

I'm reading Milnor's 'characteristic classes', and in the first chapter he defines smoothness structure on a set M, which is confusing for me, as what follows:(these are not Milnor's words so please ...
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1answer
247 views

Prove not a violation of Stokes theorem

The question is as follows: Define the vector field ${\bf F}$ on the complement of the $z$-axis by $${\bf F}(x,y,z)= \frac{-y{\bf i} +x{\bf j}}{x^{2}+y^{2}}.$$ i) Show that ...
4
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1answer
212 views

Stiefel-Whitney numbers for product bundle

I'm reading Milnor's "characteristic classes" and I want to compute Stiefel-Whitney numbers of $ P^2 \times P^2 $ (product of projective spaces) for one of the problems, I know how Stiefel-Whitney ...
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1answer
246 views

Different definitions of handle attachment

This is an extremely technical question about handle attachments.  I am asking why two definitions are equivalent.  My question appears in the second to last paragraph after I've described the two ...
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1answer
391 views

Proof that cone not diffeomorphic to plane

What is the simplest way to show that the cone $\{(x,y,z)\in\mathbb R^3\,|\,z=\sqrt{x^2 + y^2}, z\geq 0\}$ is not diffeomorphic to $\mathbb R^2$? After some comments, I realize that this question The ...
5
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2answers
96 views

Does a diffeomorphism between the interiors of two manifolds extend to the manifolds?

Let $M,N$ be manifolds with boundary. Let $f:\mathring{M}\to \mathring{N}$ be a diffeomorphism of their interiors. Does it extend to a diffeomorphism $M\to N$? I suppose we can look at the problem ...
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1answer
177 views

Extending a $C^2$-function from a $C^{1,1}$-curve to some neighbourhood

Suppose I have a simple, compact $C^{1,1}$-curve $L$ in $\mathbb{R}^3$ and a $C^2$-function $f$ on it ($C^2$ meaning with two continuous arclength derivatives). Can it be extended to a $C^2$-function ...
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1answer
205 views

Torus with positive sectional curvature.

There was this question, whether a torus in dimension n, $T^n$, can carry a riemannian metric with positive sectional curvature. A read a proof, which goes as follows: $T^n$ is complete, because ...
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Is this a trivial Stokes exercise?

I'm working on this exercise from an old Spivak Differential Geometry book which I must be misunderstanding. The question reads: Let $M$ and $N$ be compact $n$-dimensional manifolds with boundary and ...
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1answer
193 views

Immersing punctured torus

I am looking for a proof (as elementary as possible) of the fact that punctured n-dimensional torus admits an immersion to $\mathbb{R}^n$. The 2-dimensional case seems to be evident, but I haven't got ...
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67 views

Show that $\Gamma(TM) \cong_{\mathbb{R}} \mathsf{Der}_{\mathbb{R}} C^{\infty}(M)$

Let $M$ be a smooth real manifold. I want to show that we have an isomorphism of real vector space $\Gamma(TM)$ of all smooth sections of $TM$ (i.e. of vector fields on $M$) and of real vector space ...
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3answers
61 views

Lebesgue Measure with given function?

Suppose $E$ subset $R$ ($R $\is real numbers) where $E$ is Lebesgue measurable, and $f:E\to R$ and defined $g: R\to R$ by \begin{equation*} g(x) = \begin{cases} f(x) & x \in E \\ 0 & x ...
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1answer
48 views

set of immersions on a manifold is an open set in the set of all mappings to $\mathbb{R}^{2m+1}$

I'm studying a proof of Takens' Delay Embedding Theorem. A key fact used is that the set of immersions is open in the set of all mappings (mappings from an $m$-dimensional manifold M to ...
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1answer
62 views

Fundamental group of the following disc

What is the fundamental group of the following space in $\mathbf C^n$? This is the topological space given by $$\{(x_1,\ldots,x_n)\in \mathbf C^n-\{0\} : \vert x_1\vert < 1, \ldots, \vert x_n\vert ...