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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.

7287 questions
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Proving that a form is exact

Maybe this question is rather obvious but I didn't manage to solve it myself. Assume $M$ is a closed, oriented manifold. take $$ \Omega^k(M)\ni \omega = \begin{cases} d\beta~,~~~ in~ U\\ 0~,~~~ otherwise \end{cases} $$ Where $U\subset M$ is open,…
GreatGuest
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Understanding topological modular forms

I am asking for good references to understanding topological modular forms. Please don't laugh. I am more or less an analyst and differential geometer, and I do know some algebraic topology and very little algebraic geometry. Most stuff I could find…
Kofi
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homotopying a function to itself by a regular homotopy

I wonder whether there is a smooth function $f:\mathbb{S}^{1}\rightarrow \mathbb{R}^{1}$ which may be homotoped to itself by a regular homotopy $H(x,t)$, i.e. by a smooth $H:\mathbb{S}^{1}\times\lbrack0,1]\rightarrow \mathbb{R}^{1}$ such that…
sSs01
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Linking number of a pair of circles

Definition Given disjoint manifolds $M$, $N\subset \Bbb{R}^{k+1}$, the linking map $\lambda: M\times N\to S^k$ is defined by $\lambda(p, q) = (p - q)/||p - q||$. If $M$ and $N$ are compact, oriented, and boundaryless, with total dimension $m + n =…
jamal
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Example of a map which is not a diffeomorphism

Can anyone think of a bijective smooth map from a compact space to a huasdorff space which is not a diffeomorphism? thanks
kingkongdonutguy
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Finding the critical points of a map on the torus

Consider the $2$-torus $T$ obtained by revolving about the $z$-axis the circle $(x-2)^2+z^2=1$. I want to find the critical points of the map $f:T\to\Bbb{R}$ defined by $f(x,y,z)=x$. The equation of $T$ is given by $(r-2)^2+z^2=1$ where…
Xena
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coordinate system of a sphere

I am looking for a coordinate system for the sphere that has constant Lamé parameters. In fact, the Lamé coefficients of the usual spherical coordinate system are: $L_1 = R$ $L_2 = R\sin(\phi)$ As you can see, $L_2$ is linked to $\phi$. here is…
catta
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Proof of Lemma in "Differentiable Viewpoint"

On page 11 of Milnor's Differential Topology book there is Lemma 1. In the proof of Lemma 1 it says, to define, $ F:M\to N\times \mathbb{R}^{m-n}$ by $F(\xi) = (f(\xi),L(\xi))$. The derivative produces $dF_x(v) = (df_x(v),L(v))$, so far this is all…
Nicolas Bourbaki
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Show $\exists p \notin f(X) \cup Z$ by Sard

Guillemin & Pallack P83, Ex 2.4.9: Suppose $X$ compact with $0< \dim(X) < k$ and $f: X \to S^k$. Suppose $Z \subset S^k$ a closed submanifold with $\dim(X) + \dim(Z) = k.$ Show that $I_2(X, Z) = 0.$ The hint says by Sard, $\exists p \notin f(X)…
1LiterTears
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Does 2 manifolds can be "isotoped away"?

Let $M,N\subset P$ be two manifolds such that dim($M$) + dim($N$) < dim($P$), suppose that $M$ is compact and $N$ is closed, is it true that there exists an isotopy $F$ of $M$ such that $F(M,1)\cap N =\emptyset$? I already proved that we can suppose…
ADR
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Prove the directional derivative operators at a point on manifold form a vector space

One of the way to define tangent space is to use directional derivative. However, it's not clear at the first glance that the directional derivative operators form a vector space. Let $D$ be the set of all directional derivative operators at $p \in…
user110373
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Rotating about the z-axis defines a smooth function on $S^{2}$

I want to show that rotating about the z-axis defines a smooth function on $S^{2}$. To do this I used the function: $f(x,y,z)=(x\cos(\theta)-y\sin(\theta),x\sin(\theta)+y\cos(\theta),z)$ where $\theta$ is the rotation angle. This function is smooth.…
user99163
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does composition of maps is smooth and one map is smooth imply the other is also smooth?

If $f\circ g$ is smooth and $f$ is smooth, does it follow that $g$ is smooth? Note that I cannot simply take the inverse of $f$. Do I have to use implicit function theorem?
D. Huang
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Showing induced orientation on boundary is an orientation

I've found similar questions, but their accepted answers either use forms or don't quite answer what I'm looking for. To make clear the definitions I'm allowed to use, an orientation on an $m$-manifold $M$ is a choice of an orientation on each…
Dalop
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Degree mod 2 of composition is product of degree mod 2

I'm using Milnor's definition of degree mod 2 for a smooth map. Specifically, let $f: M \rightarrow N$ be a smooth map with $M$ compact without boundary, $N$ connected, and $M$ and $N$ having the same dimension. Let $y \in N$ be a regular value for…
Dalop
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