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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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Surfaces, vector fields, and the Lie bracket.

$\textbf{Theorem:}$ Suppose that $X_1, \dots, X_k$ are vector field on a manifold $M$ and at a point $p \in M$, we have that $X_1(p), \dots, X_k(p) \in T_p(M)$ are linearly independent, then the Lie brackets, $$[X_i, X_j]$$ vanish in a neighborhood…
Yuugi
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Question about proof on $F$-related vector fields in Lee

$\textbf{Proposition:}$ Suppose $F: M \to N$ is a smooth map between manifolds with out without boundary, let $X \in \mathcal{X}(M)$ and $Y \in \mathcal{Y}(N)$. Then $X$ and $Y$ are $F$-related if and only if for every smooth valued function $f$…
Yuugi
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Regular points on sphere.

I am reading Milnor's book "Topology from the Differentiable Viewpoint" and have a question of regular values. So, he defines a regular point as follows. Let $f :M \to N $ be a smooth map between smooth manifolds of the same dimension. A point $p…
Rene Cabrera
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What is the derivative of the map $\mathbb{R}^\ell\times S\to\mathbb{R}^N$ given by $(t_1,\dots,t_\ell,v_1,\dots,v_\ell)\mapsto \sum t_iv_i$?

In problem 7, pg. 75 of Guillemin and Pollack's Differential topology, there is a hint saying The set $S\subset(\mathbb{R}^N)^\ell$ of linearly independent $\ell$-tuples in $\mathbb{R}^N$ is open in $\mathbb{R}^{N\ell}$ and the map…
Clara
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Prove that in the half plane $\{x>0\}$, ω is the differential of a function.

Define a 1-form $ω$ on the punctured plane $R^2-\{0\}$ by $ω(x,y)=\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2 }dy $ a) Calculate $∫_Cω$ for any circle C of radius r around the origin b) Prove that in the half plane $\{x>0\}$, ω is the differential of a…
Diane Vanderwaif
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Transversality through two functions $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ for $W\subset Z$

For Exercise 5 section 5 chapter 2 of Guillemin & Pollak: Set $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ and assume that g is transversal to a submanifold $W\subset Z$. Show $f\pitchfork g^{-1}(W)$ iff $g\circ f\pitchfork W$. First for $f(x)=y\in…
Donyarley
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Show that at a zero $x$, the derivative $d(\vec v_x ) :T_x (X)→R^N$ actually carries $T_x (X)$ into itself.

Show that at a zero $x$, the derivative $d(\vec v_x ) :T_x (X)→R^N$ actually carries $T_x (X)$ into itself. I know that we have the vector field $\vec v:X\to R^n$. If $X=R^n \times \{0\}$, then the claim is true obviously.. The book tells me to…
XiaoXiao Zhen
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Restricting a differentiable function to a submanifold.

If $f: M \longrightarrow N$ is a differentiable function between manifolds and $A$ is a submanifold of $M$, can I conclude that $f_{|_A}$ is a differentiable function? It seems that the answer should be "Yes." (we are restricting a differentiable…
ted
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Show that $R^n -X$ has at most 2 connected component without using the Jordan-Brouwer theorem.

Here is what I got but my professor say it's wrong Let $X$ be a compact, connected hyperspace in $R^n$, then $ R^n-X$ consist of 2 open sets $D_0$ – the outside component and $D_1$ the inside component. Moreover, $∂(\overline {D_1} )=X$. Let…
XiaoXiao Zhen
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Prove that the index of a vector field is well-defined.

Consider first an open set $U \subset \Bbb{R^m}$ and a smooth vector field $v : U\to\Bbb{R^m}$ with an isolated zero at the point $z \in U$. The function $\overline{v}(x) = v(x)/\|v(x)\|$ maps a small sphere centered at $z$ into the unit sphere. The…
user67803
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Prove that $f$ map a neighborhood of $Z$ diffeomorphically on to a neighborhood of $f(Z)$( more detail)

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…
XiaoXiao Zhen
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The index of a smooth vector field is well-defined

How can I prove that the index of a smooth vector field is well-defined? All I know that it is locally constant. Definition. Given an open set $U\subset\Bbb{R^m}$, and a smooth vector field $v:U\to\Bbb{R^m}$ with an isolated zero $z\in U$, the…
algebraically_speaking
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Prove isomorphism of fundamental groups

Hei, guys! I'm having some problem solving the next exercise: Let $f: M -> N$ be a homeomorphism. Define a map $f*:π_1 (M, x_0) → π_1 (N, f(x_0 ))$ such that $f*([\gamma])=[f∘\gamma]$. Show that $f*$ is an isomorphism. Check that the map is…
Chan
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Canonical projection on submanifold $M^k $ over a hyperplane $H^{n}$ is immersion

Let $M^k \subset \mathbb{R}^{n+1}$, $M$ compact and $2k\leq n$. Show tha exist a $n$-hyperplane $H^n\subset \mathbb{R}^{n+1}$ such that if $\pi:H^{n}\oplus (H^n)^{\perp}\rightarrow H^{n}$ is the projection then $\pi\vert_{M}:M\rightarrow H^{n}$ is…
Donyarley
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Transversality, mod 2 degree, Winding numbers in differential topology

From Chapter 2 Section 5 of Guillemin and Pollack, Differential Topology, $\mathbf{X}$ is a compact connected manifold, and $f:\mathbf{X}\rightarrow \mathbb{R}^n$ a smooth map and $\dim{X}=n-1$. (So, $f$ might be the inclusion map of a hypersurface…
MathsStudent
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