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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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
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 = ...
2
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1answer
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 ...
1
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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 ...
8
votes
3answers
481 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
208 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
218 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
99 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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2answers
67 views

$f: X \to Y$ 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 ...
5
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2answers
477 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: ...
1
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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 ...
0
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1answer
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, ...
4
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1answer
254 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 ...
2
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0answers
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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0answers
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 ...
2
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1answer
174 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
149 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.
3
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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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2answers
563 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
254 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 ...
2
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1answer
616 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
958 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 ...
1
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1answer
45 views

Question on Computation of Integral of a Form

Again: I'm trying to understand the result of a certain integral of a form in a paper I'm reading (for which I do not, unfortunately, have a link): We start with a surface S that is oriented, ...
2
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2answers
383 views

A map without fixed points - two wrong approaches

For the unit sphere $S^n \subset \mathbb{R}^{n+1}$ let $f : S^n \to S^n$ be the map reversing the signs of all but one coordinate, $$f(x_0, x_1, \dots, x_n) = (x_0, -x_1, \dots, -x_n):$$ (a) ...
4
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1answer
174 views

Closed and exact.

I tried this question, but I have no idea if I got it correctly. On $\mathbb{R}^2$, let $\omega = (\sin^4 \pi x + \sin^2 \pi(x + y))dx - \cos^2 \pi(x + y)dy$. Let $\eta$ be the unique $1$-form on ...
2
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0answers
65 views

Kirby diagrams for nonorientable $4$-manifolds

In http://www.math.msu.edu/~akbulut/papers/akbulut.lec.pdf, which is a (still developed) set of lecture notes on 4-manifolds by Selman Akbulut, in section 1.5 there is a way to draw a non-orientable ...
0
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1answer
122 views

Homeomorphism under subspace topology in Hausdorff space

Let $Y$ be a Hausdorff space, and $U,V \subset Y$ are homeomorphic under subspace topology. Does this imply if $U$ is open(or closed) then $V$ is open(or closed) under original topology? I can't ...
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1answer
191 views

Lefschetz number

For the unit sphere $S^n \subset \mathbb{R}^{n+1}$ let $f : S^n \to S^n$ be the map reversing the signs of all but one coordinate, $$f(x_0, x_1, \dots, x_n) = (x_0, -x_1, \dots, -x_n):$$ Compute the ...
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1answer
89 views

Prove that $\int_C \eta \neq 0$.

I hope I write down (a) correctly. For (b), I followed Amitesh Datta's suggestion, but I hope I well-justified my argument - did I? On $\mathbb{R}^2$, let $\omega = (\sin^4 \pi x + \sin^2 \pi(x + ...
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1answer
64 views

Prove $I(Z_1, Z_2) = 0$.

Define $\mathbb{R}P^k$ as the quotient of $S^k$ by the antipodal map, with smooth structure defined so that the projection $p: S^k \to \mathbb{R}P^k$ is a local diffeomorphism. Suppose that $k$ ...
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1answer
136 views

Induced orientation on boundary

I am trying to understand how the orientation is induced on the boundary of a differentiable manifold with boundary. Here is what I have worked out so far: Let $M$ be a differentiable manifold and ...
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2answers
163 views

The preimage of $\triangle$ is a compact zero-dimensional manifold.

$f: X \to Y$, $g: Z \to Y$ and $Z$ are appropriate for intersection theory ($X,Y,Z$ are boundaryless oriented manifolds, $X,Z$ is compact, $Z$ is closed submanifold of $Y$, and $\dim X + \dim Z = \dim ...
0
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1answer
196 views

Subtraction for linear space.

I believe that for a linear space (assume finite first, but I'd like to hear about the case for infinite dimensional space.) $$V_1 + V_2 = V_3$$ does not imply that $$V_1 = V_3 - V_2.$$ Simply ...
4
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1answer
72 views

On framed manifolds

Let $M$ be an $n$-dimensional compact manifold without boundary sitting in $\mathbb R^{n+k}$. We call $M$ framed if at every point $x \in M$ there exist linearly independent vectors $v_1, \dots, v_k$ ...
5
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2answers
70 views

$T: \mathbb{R}^n\to \mathbb{R}^m$, $n>m$, $\operatorname{rank}(dT)=m$, show $T$ maps open sets to open sets.

Suppose $T: \mathbb{R}^n\to \mathbb{R}^m$, $n>m$, with $dT$ having rank $m$ at all points in an open set $D \subset \mathbb{R}^n$. What is a proof that $T$ maps $D$ into an open set in ...
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1answer
41 views

Prove $I(Z,X) = -I(X,Z)$.

I want to show that $I(Z,X) = -I(X,Z)$. So clearly I have two orientations for $X$ and $Z$ each. Do I discuss these four cases, each consider $I(Z,X)$ and $I(X,Z)$? Is this the correct approach and ...
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1answer
78 views

Plus or minus? Is there a canonical orientation, like counterclockwisely?

When $X$ also happens to be a submanifold of $Y$, then, as in the mod $2$ case, we define its intersection number with $Z, I(X, Z)$, to be the intersection num­ber of the inclusion map of $X$ with ...
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1answer
48 views

Why the degree of $p/|p|$ is zero on $\partial W^\prime?$

Let $W$ be a smooth compact region in $\mathbb{C}$ whose boundary con­tains no zeros of the polynomial $p$. $p$ has only finitely many roots, $z_0, \dots, z_n$ in $W$. Around each $z_i$, ...
3
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1answer
61 views

Why not extending to the whole disk implies have a zero

For any complex polynomial $p(z)$ of order $m$, we showed earlier that on a circle $S$ of sufficiently large radius $r$ in the plane, $$\frac{p(z)}{|p(z)|}\quad \text{and}\quad ...
3
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3answers
135 views

Inner product of De Rham cohomology classes

Is there a well-defined inner product between cohomology classes? In particular, is it possible to extend the Hodge inner product? If I try, I obtain this: $$\int *(\omega + d\lambda)\wedge (\sigma + ...
4
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1answer
164 views

What do we mean when we say a differential form “descends to the quotient”?

Let $S$ be a surface and let $f:S\rightarrow S$ be a diffeomorphism. We define the mapping torus $M_f$ of the pair $(S,f)$ to be the quotient $$(S\times I) /\sim \quad \text{ where } \ (1,x) \sim ...
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1answer
71 views

Proof that $\operatorname{rank}(dT)=1$ implies the image is a curve

I have a question about the proof that if the differential $dT$ of a transformation has rank 1 (2) at each point in a domain, then the image will be a curve (surface). Stated more precisely (in ...
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1answer
82 views

Prove $z \to z^m$ has degree $m$.

I am hoping to prove this obeying author's intention - following his hint. But I am wondering if I shouldn't employ Euler's Formula, and should use a more primitive method? I also granted my proof ...
3
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1answer
107 views

Prove $G = \left\{ \mathrm{diag} (e^{ti}, e^{\lambda ti}) \mid t \in \mathbb{R} \right\}$ is not a manifold.

Let $\lambda$ be an irrational number. Let $G \subset G_2(\mathbb{C})$ be defined as $G = \left\{ \mathrm{diag} (e^{ti}, e^{\lambda ti}) \mid t \in \mathbb{R} \right\}$. Prove that $G$ is not a ...
3
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1answer
60 views

Calculate $\deg(f)$

According to Guillemin and Pollack, Differential Topology Page 109, $f: X \to Y$ are appropriate for intersection theory ($X,Y$ are boundaryless oriented manifolds, $X$ is compact), when $Y$ is ...
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1answer
70 views

Does “Add up” just means oriented counterclockwisely?

$f: X \to Y$ and $Z$ are appropriate for intersection theory $X,Y,Z$ are boundaryless oriented manifolds, $X$ is compact, $Z$ is closed submanifold of $Y$, and $\dim X + \dim Z = \dim Y$), $f$ is ...