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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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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163 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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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
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 ...
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161 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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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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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 ...
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}) = ...
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1answer
405 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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138 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
382 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. ...
5
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1answer
193 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 ...
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310 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
289 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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43 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
206 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
120 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
129 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
57 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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74 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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478 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
207 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 ...
2
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2answers
217 views

$f^*(w \wedge \theta) = (f^*w) \wedge (f^* \theta)$

I am reading Guillemin and Pollack's Differential Topology. For the proof on Page 164, I was not able to get through the last step. $$f^*(w \wedge \theta) = (f^*w) \wedge (f^* \theta)$$ ...
2
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1answer
98 views

Pullback expanded form.

Definition. 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: $$f^*\omega(x) = (df_x)^*\omega[f(x)].$$ According to Daniel ...
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$f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, what is $\omega[f(x)]$?

I am reading Guillemin and Pollack's Differential Topology Page 163: 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: ...
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1answer
87 views

How to show the smooth map $f : T^2 \to S^3$ is an orientation-preserving diffeomorphism?

Let $f : T^2 \to S^3$ be the smooth map of a 2-torus into $S^3$. Let $\omega$ be a closed 2-form on $S^3$. Show that $$\int_{T^2}f^*\omega = 0.$$ So apparently, if I can use the theorem on ...
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472 views

Definition of pullback.

I am reading Guillemin and Pollack's Differential Topology Page 163: 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: ...
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1answer
87 views

Fill in the hole for the proof for $f^*(w \wedge \theta) = (f^*w) \wedge (f^* \theta)$

My proof has a hole there, wonder if anyone can help me fill it in? $$f^*(w \wedge \theta) = (f^*w) \wedge (f^* \theta)$$ By definition, $$f^*w(x) = (df_x)^*w[f(x)].$$ So I understand ...
3
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1answer
126 views

$d\phi = \sum \frac{\partial \phi}{\partial x_i}dx_i.$

Just a work out for a very tautological question that I am very uncertain about. If $\phi: X \to \mathbb{R}$ is a smooth function, $d\phi_x: T_x(X) \to \mathbb{R}$ is a linear map at each point ...
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47 views

$1$-forms of coordinate function $dx_i(z)(a_1, \dots, a_k) = a_i$.

I am reading Guillemin and Pollack's Differential Topology Page 163: The coordinate functions $x_1, \dots, x_k$ on $\mathbb{R}^k$ yield $1$-form $dx_1, \dots, dx_k$ on $\mathbb{R}^k$. Check $dx_1, ...
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1answer
251 views

Two definitions of smooth manifolds

In Milnor/Stasheff they give the definition of smooth manifold as follows (page 4): A subset $M \subset \mathbb R^A$ is a smooth manifold of dimension $n \ge 0$ if, for each $x \in M$ there exists a ...
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188 views

A differential topology problem.

I don't have a clue for question (b) at all. Can I get some help? Let $A,B \subset S^n$ be disjoint closed subsets. The Smooth Urysohn Theorem guarantees that there is a smooth $\phi: S^n \to ...
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30 views

Why the degree of the directional map is the number of times $\vec{v}$ rotates?

I know that $\operatorname{ind}_0(\vec{v})$ simply counts the number of times $\vec{v}$ rotates completely while we walk counterclockwise around the circle. However, if I follow the definition on ...
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1answer
26 views

$\vec{v}(x)/|\vec{v}(x)|$ extends to the annulus bounded by the two spheres.

Guillemin and Pollack's Differential Topology Page 133: $\vec{v}$ is a smooth map $\vec{v}: X \to \mathbb{R}^n$ such that $\forall x, \vec{v}(x) \in T_x(X)$. Assume that we are in $\mathbb{R}^k$ and ...
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1answer
168 views

Proof the degree of a reflection through a hyperplane is −1.

This seems intuitively true, but it seems impossible to work it out according to the definition on Guillemin and Pollack's Differential Topology Page 108, tracing back t0 107 and 100. Page 108: ...
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1answer
145 views

There is no immersion of the Möbius band in the plane.

There is no immersion of the Möbius band in the plane. I believe we have to work with the tangent bundle of the Möbius band, but I'm not getting no useful result.
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1answer
120 views

Covering a genus g surface with n disks, and conversely

Suppose $X$ is a closed compact surface (i.e. a compact two-dimensional manifold without boundary). If $X$ has genus $g$, what is the minimum number of (homeomorphic images of) open balls required ...
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549 views

Euler characteristic of an $n$-sphere is $1 + (-1)^n$.

I am using the textbook Guillemin and Pollack's Differential Topology but I am asked to solve a question need this fact that Euler characteristic of sphere is $1 + (-1)^n$. So may I ask if this is ...
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1answer
248 views

Preimage orientation.

On Guillemin and Pollack's Differential Topology Page 100. Let $f: X \to Y$ be a smooth map with $f \pitchfork Z$ and $\partial f \pitchfork Z$, where $X,Y,Z$ are oriented and the last two are ...
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1answer
603 views

The degree of antipodal map.

I am trying to solve the problem A map without fixed points. But I am not certain about the degree of antipodal map. I my thought, since the preimage of a point $y \in S^k$ is just $-y$, the degree ...
7
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2answers
937 views

map of arbitrary degree from compact oriented manifold into sphere

This is a question from a qualifying exam. Let $X$ be a compact, oriented $n$-dimensional manifold. Show that for any $k \in \mathbb{Z}$, there exists a continuous map $f: X \to S^n$ of degree $k$. I ...