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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Constant Rank Theorem and Submanifolds

I'm related to my previous question here. The problem is: I am given a $C^r$ manifold $M$ and a connected subset $A$ of $M$, and a $C^r$retraction $f:M\rightarrow A$ such that ...
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Is this enough to show that this map has constant rank?

My question is related to this question I am given a $C^r$ manifold $M$ and a connected subset $A$ of $M$, and a $C^r$retraction $f:M\rightarrow A$ such that $f\vert_A=id:A\rightarrow A$. Then $A$ is ...
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76 views

On the proof that the inverse value set of a regular value is a submanifold

I have a doubt on the proof of the following, well-known theorem: Let $f:M^m\rightarrow N^n$ ($m\geq n$) be a $C$ map, $r\geq 1$.If $y\in f(M)$ is a regular value, then $f^{-1}(y)$ is a $C$ ...
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Typo or additional hypotheses needed in problem 5, p.71 of Bredon's Topology and Geometry?

The problen can be foun in p.71 of Topology and Geometry. I state it below for convenience. Problem: Consider the half open real line $X=[0,\infty)$. Define a functional structure $F_{1}$ by taking ...
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193 views

Isotopy preserving inverse image $f_t^{-1}(V)$ of a homotopy

During a lecture I was given a bunch of easy propositions entitled as "observations" by the lecturer. But one of them seems more difficult and I have absolutely no idea how to "observe" it... I'd be ...
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106 views

Almost complex structures on a manifold and its exotic copy

Consider a compact simply-connected smooth $4$-manifold $M$ and its exotic copy $M'$. Identify the underlying topological manifolds. Dold-Whitney theorem guaranties ($\mathbb R$-linear) isomorphism of ...
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608 views

Immersion is a diffeomorphism.

Clearly I suppose to put the condition $n > 1$ in use. So my proof must went wrong.. Could someone help me take a look at it? Thanks! Suppose $X$ is a smooth, compact, connected $n$-manifold ...
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54 views

What is the killing field on $S^2$ and $SO[3]$?

What is the killing field on $S^2$ and $SO[3]$? I understand the structure on $S^1$, but not sure about how the vectors work on $S^2$. Thanks in advance!
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237 views

A functional structure on the graph of the absolute value function

Let $X$ be the subspace of $\mathbb{R}^2$ consisting of the graph of the absolute value function. That is, $X=\{(x,|x|) : x\in\mathbb{R})\}$. We define a functional structure on $X$ by restricting ...
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54 views

$G$ must hit the origin. - Is this proof legit?

The map $F : \mathbb{R}^3 \to \mathbb{R}^3$ given by $F(x, y, z)=(-z, -x, -y)$ restricts to a map $f : S^2 \to S^2$ from the 2-sphere to itself. Show that if $G$ is another map of Euclidean ...
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61 views

Show that the Euler characteristic of $O[3]$ is zero.

Show that the Euler characteristic of $O[3]$ is zero. Consider a non zero vector $v$ at the tangent space of identity matrix. Denote the corresponding matrix multiplication by $\phi_A$. Define ...
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388 views

When characteristic classes determine a bundle?

Let $k$ be a natural number and $F$ be $\mathbb C$ or $\mathbb R$. What conditions should be imposed on a topological space that it was true that a $k$-dimensional vector bundle defined over $F$ on ...
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2answers
277 views

A submersion $F : \mathcal{X} \to \mathcal{Y}$ must be surjective.

Let $\mathcal{X}$ and $\mathcal{Y}$ be compact manifolds and let $\mathcal{Y}$ be connected. Prove that a submersion $F : \mathcal{X} \to \mathcal{Y}$ must be surjective. I don't have much ...
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73 views

Prove that $\mathcal{M}=\{(x, y) \in X \times Y : f(x) = g(y)\}$ is a smooth submanifold.

Let $\mathcal{X}^m, \mathcal{Y}^n,\mathcal{Z}^p$ be manifolds. Let $f : \mathcal{X} \to \mathcal{Z}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be maps such that $f \pitchfork g$. Prove that ...
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401 views

Diffeomorphisms are either both orientation-preserving or both orientation reversing.

Let $F, G : M \to N$ be diffeomorphisms of compact, connected, oriented, $n$-manifolds. If $F$ and $G$ are smoothly homotopic, prove that $F$ and $G$ are either both orientation-preserving or both ...
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177 views

Tangent bundles of exotic manifolds

Consider a pair of homeomorphic but not diffeomorphic smooth manifolds $M_1$ and $M_2.$ Fix a homeomorphism $\phi\colon M_1\rightarrow M_2.$ If I understand correctly, two bundles $TM_1$ and ...
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66 views

$\int_{S^1} \beta = 0 \Rightarrow \beta$ is the differential of a function. - Is this proof legit?

Let $\beta$ be a smooth $1$-form on $S^1$, and $\int_{S^1} \beta = 0.$ Prove $\beta$ is the differential of a function. I don't really have a clue for this question.. I am trying to follow Anthony ...
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33 views

The global Lefschetz number of $f$ vanishes. - Is this conterexample work?

I am hoping someone will be willing to help me take a look at if this conterexample works? Let $X$ be an oriented compact manifold and $f : X \to X$ a map. Suppose $W$ is a compact oriented ...
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45 views

For which values of $n$ does there exist a compact, oriented 3-manifold $X$.

So I could barely understand the problem statement ("oriented boundary given by a surface $F$ having the map $f$"), nor how to proceed. Can I get some hints? Thank you. Consider the smooth map $f: ...
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88 views

Compute $d\omega$ and $\int_{S^2}\omega$.

I am wondering if my solution is correct? Thanks. (a) On $\mathbb{R}^3$, let $\omega = y dx \wedge dz.$ Compute $d\omega$ and $\int_{S^2}\omega$, where $S^2$ is the unit sphere centered at the ...
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60 views

$f$ extends to a smooth map $\tilde{f} : S^2 \to S^2$. - Is the proof legit? [duplicate]

Let $f : \{\mathbb{R}^2 - (1, 0) - (-1, 0)\} \to \mathbb{R}^2$ be the function $f(z) = \frac{1}{z-1} - \frac{1}{\bar{z}+1}$. Show that $f$ extends to a smooth map $\tilde{f} : S^2 \to S^2$. We ...
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52 views

$\forall M \in \mathrm{SL}(2, \mathbb{R})$ has a neighborhood diffeomorphic to $\mathbb{R}^3$?

$\forall M \in \mathrm{SL}(2, \mathbb{R})$ has a neighborhood diffeomorphic to $\mathbb{R}^3$? Known that $\mathrm{SL}(2, \mathbb{R})$ is a three-dimensional manifold.
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83 views

Does $\mathrm{SL}(2, \mathbb{R})$ have a non-empty boundary?

Does $\mathrm{SL}(2, \mathbb{R})$ have a non-empty boundary? Edit: I've shown that $1$ is a regular value, and hence $\mathrm{SL}(2, \mathbb{R})$ is a three-dimensional manifold as Ted's hint. ...
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107 views

The monodromy and cut planes.

I am trying to understand the following example from the book Riemann surface by Donaldson (p.48). This was given after introducing the notion of monodromy of the covering. Consider the Riemann ...
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73 views

Assume that $f: X \rightarrow Y$ is a smooth map between two smooth manifolds.

Assume that $f: X \rightarrow Y$ is a smooth map between two smooth manifolds. Must there exist a smooth manifold $Z$, a submersion $g:X \rightarrow Z$, and an immersion $h:Z \rightarrow Y$ such that ...
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269 views

Request for companion of Mariano Suárez-Alvarez's proof.

Mariano Suárez-Alvarez's answer to Cohomology of projective plane seems very interesting. However, there are three pieces I could not stitch up for one of his proofs. Wonder if someone may help? ...
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162 views

Show that $f$ extends to a smooth map.

Identify $\mathbb{R}^2$, with coordinates $x, y$, with $\mathbb{C}$, with coordinate $z = x + iy$. Likewise, identify a copy of $\mathbb{R}^2$ with coordinates $u, v$ with $\mathbb{C}$ with ...
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127 views

Sphere turned inside out

How can I turn a sphere inside out? I saw this video on YouTube and I didn't understood how can i turn a sphere inside out. any help will be appreciated. http://www.youtube.com/watch?v=R_w4HYXuo9M
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393 views

What is the difference between differential topology and calculus on manifolds?

I'm trying to teach myself one and bought a book on the other. It seems to me that they both cover about the same material. This leads to the question: What is the difference between differential ...
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87 views

The square of the Euclidean distance is smooth.

Let $S^2 \subset E^3$ be the unit $2$-sphere in Euclidean $3$-space. Set $M = \{p_1, p_2 \in S^2 : p_1 \neq p_2\}$. Define $f : M \to \mathbb{R}$ by setting $f(p_1, p_2)$ to be the square of ...
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364 views

“Coordinate functions” on the structure-sheaf definition of a smooth manifold

I've been reading Bredon's Topology and Geometry recently; what an excellent book! He defines smooth manifolds in two distinct ways and then shows they are in fact equivalent. The "non-standard" ...
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57 views

The winding number of $f$ about $0$

Let $f: S^1 \to \mathbb{R}^2 - \{0\}$ be a smooth map. Define the winding number of $f$ about 0 and prove that it equals $\frac{1}{2\pi}\int_{S^1}f^*(d\theta).$ According to the definition, we ...
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157 views

If $M$ is diffeomorphic to $N$, then $\mathbf{H}_{DR}^p(M)$ is isomorphic to $\mathbf{H}_{DR}^p(N)$.

I thought I got this, but no..... Given $\mathbf{H}_{DR}^2(S^3)$ is trivial but $\mathbf{H}_{DR}^2(T^3)$ is not, how can I show $S^3$ and $T^3$ are not diffeomorphic? I am also wondering about the ...
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123 views

form exact $\Leftrightarrow$ pull-back exact

Is the form exact $\Leftrightarrow$ pull-back exact? Since $$f^*\omega = \omega \circ df,$$ which seems irrelavant. Because the composition with $df$ does not change $\omega$ is exact or not. The ...
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289 views

How do I prove that a subset of a manifold is not a submanifold?

I know of ways to prove that a given subset of a smooth manifold is a smooth submanifold, but what if I have some subset which I suspect is not a smooth submanifold? What are some approaches to ...
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23 views

The orientation preserving of folliation.

A foliation $\mathcal{F}$ on a compact, oriented manifold $X$ is a decomposition into $1-1$ immersed oriented manifolds $Y_\alpha$ (not necessarily compact) that is locally given (preserving all ...
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117 views

Let $M$ be a smooth simply-connected compact manifold of dimension $n$. Is there an immersion of $M$ into the $n$-torus $T^n = S^1 × . . . × S^1?$

I do not know how to do the following qualifying exam problem. Any helped is nice. Let $M$ be a smooth simply-connected compact manifold of dimension $n$. Is there an immersion of $M$ into the ...
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1answer
67 views

$S^3$ and $T^3$ are not diffeomorphic.

Let $f : T^2 \to S^3$ be the smooth map of a 2-torus into $S^3$, therefore $$\int_{T^2}f^*\omega = 0.$$ There is a closed $2$-form $\beta$ on $T^3 = S^1 \times S^1 \times S^1$ and a map $g : ...
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136 views

Is volume form equal to the top-dimensional form with coefficient $1$?

Wikipedia says a volume form on a differentiable manifold is a nowhere-vanishing top-dimensionial form. And Guillemin and Pollack says $f: V \to U$ is a diffeomorphim of two open sets in ...
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315 views

Functionally structured spaces and manifolds

The question requires some definitions that I have listed below for your convenience. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups. On a ...
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268 views

Show that the set $M$ is not an Embedded submanifold

How can I prove that $M=\{(x,y)\in \mathbb{R}^2\ ; y=|x|\}$ is not an embedded smooth submanifold of $\mathbb{R}^2$?
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501 views

Example of commuting vector fields generating globally noncommuting flows

Recently, I discovered that a theorem from my differential geometry lecture is false due to too big generality - it stated that for vector fields $X,Y$ we have the equivalence: Incorrect! $[X,Y] = ...
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272 views

1- forms on a torus

I think this is a very simple question but I'm not really confident in mathematics (even if I like it very much) Let's fix a cube $[0,1]^3$ in $R^3$ and identify opposite sides, so as to construct a ...
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214 views

Is it possible to make the set $M:=\{(x,y)\in\mathbb{R}^2:y=|x|\}$ into a differentiable manifold?

1) Is it possible that be given a suitable smooth atlas to the set $M:=\{(x,y)\in\mathbb{R}^2: y=|x|\}$ so that $M$ to be a differentiable manifold? Why? How? 2) How can I prove that M is not an ...
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38 views

Embedding of the cotangent of the n sphere in R^2n

It is true that $T^* S^1$ embeds in $\mathbb{R}^2$ as the cotangent of $S^1$ is trivial and $S^1 \times \mathbb{R}$ is diffeomorphic to the punctured plane. Is it true in general that $T^* S^n$ embeds ...
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1answer
162 views

Integration formula with wedge product.

On $\mathbb{R}^3$, consider a compactly supported $2$-form $$\omega = f_1 \, dx_2 \wedge dx_3 + f_2 \, dx_3 \wedge dx_1 + f_3 \, dx_1 \wedge dx_2.$$ Choose for $S$ the parametrization $h: ...
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38 views

Written any surface as the graph of a function locally.

I see an interesting statement: Locally any surface may be written as the graph of a function, although one must sometimes write one as a function of the others. So, does this infer such case, ...
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146 views

1-form with positive integral over a path

Given any smooth path $\gamma$ on a manifold $M$, when can we construct a 1-form $\omega \in \Omega^1(M)$ so that $\int_{\gamma} \omega > 0$? It is easy to see that such $\omega$ exists if the path ...
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46 views

Extending a 2-frame field - manifolds with boundary

It is known that every smooth manifold can be homotoped to a cell complex. In particular this is true for manifolds with boundary. My question: Under the homotopy to a cell complex, is the boundary ...
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27 views

Examples of non-parametrizable sets?

Encountered the term parametrizable for the first time: The support of $\omega$ is contained inside a single parametrizable open subset $W$ of $X$. So I am just curious, what kind of sets are ...