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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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 ...
2
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1answer
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 ...
2
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1answer
417 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 ...
8
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1answer
185 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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0answers
34 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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0answers
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: ...
3
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1answer
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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2answers
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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1answer
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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2answers
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. ...
3
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0answers
109 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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0answers
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 ...
4
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1answer
285 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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1answer
164 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 ...
2
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1answer
128 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
3
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2answers
414 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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0answers
88 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 ...
4
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1answer
377 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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0answers
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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2answers
160 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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0answers
127 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 ...
2
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1answer
297 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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0answers
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 ...
3
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1answer
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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1answer
138 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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1answer
331 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 ...
2
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1answer
274 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 $\mathbb{R}^2$?
4
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1answer
509 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] = ...
5
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1answer
281 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 ...
4
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3answers
216 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 ...
4
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1answer
39 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 ...
2
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1answer
167 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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0answers
39 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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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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1answer
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 ...
0
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1answer
28 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 ...
5
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1answer
164 views

Compensation of the anticommutativity of wedge product.

In Guillemin and Pollack, Differential Topology Page 166, The automatic appearance of the compensating factor $\det (df)$ is a mechanical consequence of the anticommuntative behavior of $1$-forms: ...
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1answer
42 views

$f^*\left(\sum_I a_I \, dx_I\right) = \sum_I (f^*a_I) \, df_I$

By linearity of $f^*$, $$f^*\left(\sum_I a_I \, dx_I\right) = \sum_I f^* (a_I \, dx_I)$$ And if I want $df_I$, I would have to use the formula $f^*dx_i = df_i$. So $f^*$ disappears when introduced ...
4
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1answer
29 views

Commutativity of $Y^j$.

$U \subset \mathbb{R}^k$ and $V \subset \mathbb{R}^l$ be open subsets. Let $f: V \to U$ to smooth. Use $x_1, \dots, x_k$ for the standard coordinate functions on $\mathbb{R}^k$ and $y_1, \dots, y_l$ ...
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0answers
76 views

Vector field from Lamination.

Let $S$ be a smooth closed (i.e. compact without boundary) surface. A geodesic lamination on $S$ is a nonempty closed subset of $S$ which is a disjoint union of geodesics. Suppose $\alpha$ is a ...
2
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1answer
52 views

$ (f^*dx_i)(\frac{\partial}{\partial y^j}) = dx_i(f_*(\frac{\partial}{\partial y^j}))$

$U \subset \mathbb{R}^k$ and $V \subset \mathbb{R}^l$ be open subsets. Let $f: V \to U$ to smooth. I am wondering how can I show $$ (f^*dx_i)(\frac{\partial}{\partial y^j}) = ...
10
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1answer
411 views

Folliation and non-vanishing vector field.

The canonical foliation on $\mathbb{R}^k$ is its decomposition into parallel sheets $\{t\} \times \mathbb{R}^{k-1}$ (as oriented submanifolds). In general, a foliation $\mathcal{F}$ on a compact, ...
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3answers
142 views

Notation on the tangent space.

Consider $Y$ an element of the $n$-dimensional tangent space $T_yY$. The the canonical basis is $(\frac{\partial}{\partial y^1}, \cdots, \frac{\partial}{\partial y^n}).$ Then should I write $$Y = ...
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1answer
385 views

Thinking of giving up..

I got really stuck to the end of Guillemin and Pollack (in particular, here) and plan to give up. Give up Guillemin and Pollack, not math though. It seems John Milnor's classic little book topology ...
2
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1answer
85 views

Why can an orientation on $X$ be written as a sum of cycles on this open cover?

I'm reading a proof of the following theorem Let $X$ and $Y$ be compact connected oriented $n$-manifolds, and let $f:X\to Y$ be continuous. Let $y\in Y$, and assume that $f^{-1}(y)$ is finite. ...
5
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1answer
195 views

Topologist's Sine Curve not a regular submanifold of $\mathbb{R^2}$?

I am trying to work out the details of following example from page 101 of Tu's An Introduction to Manifolds: Example 9.3. Let $\Gamma$ be the graph of the function $f(x) = \sin(1/x)$ on the ...
2
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0answers
53 views

Problem reduced to analyzing solutions of a family of nonlinear systems of equations

This was posted on mathoverflow about two weeks ago and I got no response so I'm asking here in case anyone has any ideas. Original post is here. I was able to reduce a research problem relating to ...
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2answers
125 views

Winding number of $f$ is equal to $\frac{1}{2\pi}\int_{S^1}f^*(d\theta).$

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 $\frac{1}{2\pi}\int_{S^1}f^*(d\theta).$ I have no clue, except for the ...