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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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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45 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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Polyhedral pair $(S^1 \times S^1 \times S^1, S^1 \times S^1 \times \{1\})$
What is the simplicial complex pair $(K_1, K_2)$ such that $(|K_1|, |K_2|)$ gives the triangulation for $(S^1 \times S^1 \times S^1, S^1 \times S^1 \times \{1\})$. Can we write down all the simplexes ...
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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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58 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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Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both mobius 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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44 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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22 views
Let $X_\alpha$ be the connected components by arcs of $X$ (homology)
Let $X_{\alpha}$ the connected components by arcs of $X$ if $A\subset X$ and $A_{\alpha}=A\cap X_{\alpha}$ Then $H_{\ast}(X,A)=\bigoplus H_{\ast}(X_{\alpha},A_{\alpha})$
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70 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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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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92 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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70 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
86 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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56 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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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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69 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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59 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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34 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
47 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 ...
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27 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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39 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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70 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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47 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
67 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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Question on the exponential map [duplicate]
I need help showing that the map exp: $so(2) \rightarrow SO(2)$ is surjective. I have already figured out that $so(2) = \{A \in M(2, \mathbb{R}) \ | \ A^T + A = 0\}$ and that the map exp is given by ...
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62 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
45 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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Differential Topology Question on Proper Maps
I am having a difficult time whether or not the map $f: M(n, \mathbb{R}) \rightarrow M(n, \mathbb{R})$ given by $f(X) = AXA^{-1}$ is a proper map where $A$ is an invertible matrix.
Any help would be ...
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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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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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66 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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65 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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38 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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35 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 ...
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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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42 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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87 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 ...
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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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149 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
79 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 ...
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2answers
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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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79 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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50 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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27 views
Sections of the tensor product of two vector bundles
How can we define the Sections of the tensor product of two vector bundles ?
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75 views
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
47 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 ...
