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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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Extending a function on embedded submanifold: what techniques are to be used? (2.3, Lee's Riemannian Manifolds)

I am asking about a slightly different version of this question, where we are given an embedded submanifold $M \subset M'$ and are asked to extend any smooth function on $M$ to one on a neighborhood of $M$ in $M'$. Assuming our function to be…
Andrew Thompson
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page 4 of Milnor's book on Morse Theory

I have a stupid and probably naive question about one line in the book of Milnor about Morse theory. What does exactly means if $v \in T_pM$ then there is an associated vector field $\tilde v $ ? I have a kind of vague idea of what could be this…
user48483
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What does it mean for two submanifolds to have contact of order $k$?

Let $M$ be a smooth manifold of dimension $(m+n)$. Two curves $\gamma_1, \gamma_2 \colon \mathbf{R} \to M$ with $\gamma(0) = p$ are said to have contact at $p$ of order $k$ if for all smooth maps $\varphi \colon U \to R$ where $U \subset M$ is open…
Andrew Thompson
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If $M$ is a manifold and $\dim M\geq n-2$, is $\mathbb{R}^n\setminus M$ never connected and simply connected?

Suppose $M$ is a smooth submanifold of $\mathbb{R}^n$. Through some transversality tricks, I was able to prove that if $\dim M
Clara
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Transversality of a function to a sphere

I'm working through problem 6-9 in Lee Smooth Manifolds and I'm stuck. Let $F: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ defined by $(x,y) \mapsto (e^{y}\cos(x), e^{y}\sin(x), e^{-y})$. For which positive values of $r$ is $F$ transverse to the sphere…
user220613
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Prove that any map $f\colon S^1 \to S^1$ mapping antipodal point to antipodal point has $\deg_2(f)=1$ by a direct computation.

Without using the Borsuk-Ulam theorem. Prove that any map $f\colon S^1 \to S^1$ mapping antipodal point to antipodal point has $\deg_2(f)=1$ by a direct computation. I know that $f$ map anitpodal point to antipodal point, meaning $f$ is an odd…
XiaoXiao Zhen
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Show that there exist a compact manifold with boundary $W$ in $Y \times I$ such that $\partial W= X\times \{0\} \cup Z \times \{1\}$

I found this question had been posted by someone, but got no answer, so I hope I will have better luck. Show that if $X$ may be deformed into $Z$ then $X$ and $Z$ are cobordant. Deformation definition: deformation of a submanifold $Z$ in $Y$ is a…
Diane Vanderwaif
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Prove that $f$ map a neighborhood of $Z$ diffeomorphically on to a neighborhood of $f(Z)$

Supposed that the derivative of $f:X\to Y$ is an isomorphism whenever $x$ lies in the sub-manifold $Z \subset X$, and assume that $f$ maps $Z$ diffeomorphically onto a $f(Z)$ . Prove that $f$ map a neighborhood of $Z$ diffeomorphically on to a…
Diane Vanderwaif
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Prove that $f$ is Morse function if an only if $det(H)^2 + \sum_{i=1}^k (\frac {\partial f}{\partial x_i})^2>0$

Let $f$ be a smooth function on an open set $U\subset R^k$. For each $x \in U$ let $H(x)$ be the Hessian Matrix of $f$, whether $x$ is critical point or not. Prove that $f$ is Morse function if an only if $$det(H)^2 + \sum_{i=1}^k (\frac {\partial…
Diane Vanderwaif
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If two Curves in $\Bbb R^3$ are transversal then they do not intersect

Proving this is easy: Let $X$ and $Z$ be the two curves in $\Bbb R^3$. Assume $X$ and $Z$ intersect at a point $y$. Then, at most $\dim(T_y(X) + T_y(Z)) = 2$, where $T_x(X)$ and $T_z(Z)$ denotes the tangent space of $X$ at $x$ and $Z$ at $z$,…
MathsStudent
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Equality between support for a function and closed union of elements of a partition of unity (Proof from John Lee's Smooth Manifolds)?

I have a minor question in the following proof from John Lee's Intro to Smooth Manifolds: At the end, there is the equality $\mathrm{supp}\tilde{f}=\overline{\bigcup_{p\in A}\mathrm{supp}\psi_p}$. It's clear that $\tilde{f}(x)\neq 0$ requires that…
Clara
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Does there exist a dipole field on $S^2$ differing by at most a minus sign between antipodal points?

Consider the two-sphere $S^2 \subset \mathbb{R}^3$. By a dipole field on $S^2$, I mean a continuous function $f \colon S^2 \to S^2$ such that (1) $x$ is perpendicular to $f(x)$ for all $x \in S^2$ (this means that $f$ is a continuous tangent vector…
Doug
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Prove that it is a regular point

Good night, my friends, can you help me with these exercise? Let $N$ a $k$-manifold, $X$ a compact $(k+1)$-manifold in $\mathbb R^N$, and $F:X\rightarrow N$ a differentiable map. Let $y\in \operatorname{Reg}(F)\cap \operatorname{Reg}(F|_{\partial…
Framate
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A consequence of simply connectedness

Is it true that if a manifold is simply connected then its tangent bundle is simply connected?
cars
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Tangent space to the graph of a function

$X$ and $Y$ are smooth manifolds and let $f:X\to Y$ be a map. Let $\Gamma$ be the graph of $f$ in $X\times Y$. Prove that $T_{(x,y)}\Gamma$ is the graph of $df(x):T_xX\to T_yY$. $\Gamma$ need not be a smooth manifold, does it? If it were I could…
bluebox
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