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Questions tagged [differential-topology]

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.

Differential topology is the field dealing with differentiable functions on differentiable manifolds. Somewhat simplified view of this field is that it describes the setting to which the notion of differentiable function can be generalized from the more familiar case of functions $\mathbb R^n\to\mathbb R^k$. Differential topology is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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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 = \int_V (a \circ f) \left|\det(df)\right|dy_1 \cdots…
Tumbleweed
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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 $p$-form $f^*\omega$ on $X$ as follows:…
Tumbleweed
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Linearity of everything

May I ask for details about how can I prove "linearity of everything" for the following step? $(f^*dx_i)(Y) = \sum_{j = 1}^lY^j (f^*dx_i)(\frac{\partial}{\partial y^j}) = \sum_{j = 1}^lY^jdx_i(f_*(\frac{\partial}{\partial y^j}))$.
Tumbleweed
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$f^*(w_1 + w_2) = f^*w_1 + f^* w_2$

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 alternating $p$-tensor. Prove $f^*(w_1 + w_2) = f^*w_1 +…
WishingFish
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Show that $S^n$ is not contractible in $S^n \times S^n$

Show that $\{x\} \times S^n \hookrightarrow S^n \times S^n$ is not contractible It is correct to say that $\{x\} \times S^n$ is diffeomorphic to $S^n$ then it would be enough to prove that $S^n$ is contractible in $S^n \times S^n$ intuitively I…
Nick_W
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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: $$f^*\omega(x) = (df_x)^*\omega[f(x)].$$ So my question is,…
WishingFish
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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 Guillemin and Pollack's Differential Topology Page…
WishingFish
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$\vec{v}(x)/|\vec{v}(x)|$ 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 that $\vec{v}$ has an isolated zero at the origin.…
WishingFish
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The set of closed maps is closed, but not open in $C_S^\infty(\mathbb{R}, \mathbb{R})$.

The set of closed maps is closed, but not open in $C_S^\infty(\mathbb{R}, \mathbb{R})$. (A map $f$ is closed if it takes closed sets onto closed sets.) $C_S^r(M, N)$ is the set of $C^r$ maps who's topology is defined by the strong basic neighborhood…
Math_Day
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Lie Bracket of polar coordinate vector fields.

I've been thinking about this problem for some time, yet I am unable to solve it or find a resource online. I know that $\frac{\partial}{\partial r}$ and $\frac{\partial}{\partial \theta}$ are coordinate vector fields and their bracket should be…
H. M. El-Hasan
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Hopf invariant of composition of functions

Given two functions $f: S^{2p-1} \to S^p$ and $g: S^{p} \to S^{p}$, I want to prove that \begin{equation} H(f) = (\text{deg}(g))^2 H(f), \end{equation} where $H(f)$ is the Hopf invariant of $f$. I first tried to analyze the problem by assuming that…
Cal22
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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 Lefschetz number $L(f)$. So I am actually…
WishingFish
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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 $\omega = (\sin^4 \pi x + \sin^2 \pi(x + y))dx - \cos^2 \pi(x + y)dy$. (a) Show that $\omega$…
WishingFish
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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 $\dim X + \dim Z = \dim Y$), $f$ is transversal to $Z$. (1) If…
1LiterTears
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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 $Z$. If $X \pitchfork Z$, then $I(X, Z)$ is…
1LiterTears
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