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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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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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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289 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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165 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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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
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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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90 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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393 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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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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131 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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1answer
304 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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1answer
121 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
68 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
142 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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334 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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1answer
276 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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1answer
520 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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1answer
287 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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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 ...
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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 ...
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172 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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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 ...
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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 ...
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166 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 ...
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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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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 ...
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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}) = ...
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421 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
145 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
393 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 ...
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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. ...
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1answer
198 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 ...
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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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126 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 ...
2
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1answer
331 views

The Generalized Stokes Theorem.

The Generalized Stokes Theorem. If $\omega$ is any smooth $(k-1)$ form on $X$, then $$\int_{\partial X} \omega = \int_X d\omega.$$ Let $C \subset \mathbb{R}^2$ be a (smooth) simple closed ...
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1answer
301 views

Change of Variable vs. Change of Coordinates.

Are they the same thing? So given an example, I could work out by change of coordinates, but how can I apply Change of Variable to replace this process? Change of Variable in $\mathbb{R^k}$. ...
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1answer
45 views

Abbreviation of volumn form

Change of Variable in $\mathbb{R}^k$. Assume that $f: V \to U$ is a diffeomorphism of open sets in $\mathbb{R^k}$ and $a$ is an integrable function on $U$. Then $$\int_U a dx_1 \cdots dx_k = ...
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Textbooks with exposition done mostly in proof outlines or exercises?

As the title indicates, I'm trying to find books where the exposition of the main course of thought is done entirely or mostly in outlines of proofs, or as exercises with or without hints. I'm trying ...
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1answer
217 views

Define pull-back on a manifold by pull-back on the linear space.

It appears to me that pull-back on a manifold If $f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, define a $p$-form $f^*\omega$ on $X$ as follows: $$f^*\omega(x) = ...
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2answers
121 views

Linearity of push forward $F_*$

How can I prove the linearity of $F_*$? What does $F_*$ eat? If $N$ is smooth manifolds and $F: M \to N$ is a smooth map, for each $p \in M$ we define a map $F_*: T_pM \to T_{F(p)}N$, called the ...
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1answer
130 views

Linearity of $f_*, f^*$.

The definition of $f^*$ is given to me as below. But what is $f_*$? How can I justify $f_*, f^*$ is linear? Definition. If $f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, define a ...
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1answer
58 views

Linearity of everything

May I ask for details about how can I prove "linearity of everything" for the following step? $(f^*dx_i)(Y) = \sum_{j = 1}^lY^j (f^*dx_i)(\frac{\partial}{\partial y^j}) = \sum_{j = ...
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1answer
75 views

$(f\circ h)^* \omega = h^*f^*\omega$ - Legit now?

Three pullback identities are required to prove on Guillemin and Pollack's Differential Topology on Page 164. I am not sure if I get it since it is the first time I play with them. I hope I got the ...
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1answer
48 views

$f^*(w_1 + w_2) = f^*w_1 + f^* w_2$

A few pullback identities are required to prove on Guillemin and Pollack's Differential Topology on Page 164. I am not sure if I get it since it is the first time I play with them. $w$ is an ...
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485 views

$f^*dx_i = \sum_{j=1}^l \frac{\partial f_i}{\partial y_j} dy_j = df_i$

Guillemin and Pollack's Differential Topology Page 164: $U \subset \mathbb{R}^k$ 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 ...
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1answer
216 views

About zeros of vector fields in compact surfaces

I'm studying compact surfaces and in particular the relationship between zeros of vector fields defined on them and Euler characteristic of the surface herself. Let be $S$ a compact (smooth) surface ...