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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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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 $w[f(x)]$ as the $p$-tensor eats a $p$-vector $f(x)$. The…
WishingFish
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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 $\phi: S^n \to [0,1]$ with $\phi|_A \equiv 0$ and $\phi|_B \equiv…
WishingFish
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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 boundaryless. We define a preimage orientation on the…
WishingFish
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Smooth functions $\mathbb{S}^1 \rightarrow \mathbb{R}^2$

I'm reading some lecture notes on differential topology, and it is said that any smooth function $\mathbb{S}^1 \rightarrow \mathbb{R}^2$ whose image is the figure "8" is an immersion but not an embedding. I tried to prove it on my own, but I…
John Mars
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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 below is correct? Prove $z \to z^m$ has degree $m$.…
WishingFish
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How to show $\dim T_x(X) = \dim df_x T_x(X).$

$f: X \to Y$ 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 transversal to $Z$. According to the text: If…
1LiterTears
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How does one define mixed partial derivatives for functions on a manifold?

Suppose $f: M \rightarrow \mathbb{R}$ is a smooth function on a manifold $M$ and $v \in T_pM$ a tangent vector at the point $p$. First let me recapitulate the definition of $df(v)$. Suppose $\gamma : (-\epsilon, \epsilon) \rightarrow M$ is a…
Ritwik
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Does $S^1$ has boundary?

According to Guillemin and Pollack's Differential Topology: The sum of the orientation numbers at the boundary points of any compact oriented one-manifold $X$ with boundary is zero. By The Classification of One-manifold, every compact, connected,…
1LiterTears
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How a compact interval $X = [0,1]$ inherite standard orientation from $\mathbb{R}$.

According to Guillemin and Pollack's Differential Topology: A compact interval $X = [0,1]$ inherite standard orientation from $\mathbb{R}$. To my understanding, "inherit from $\mathbb{R}$" means a linear transformation between $X$ and the standard…
1LiterTears
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Why orientation on $S^1$ is the one that counter-clockwise-pointing vectors are positive.

I am totally disoriented by orientation.. So in the text, it says The closed unit vall $B^2$ in $\mathbb{R}^2$ inherits the standard orientation from $\mathbb{R}^2$. The induced orientation on $S^1$ is the one for which the…
1LiterTears
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Need help understanding orientation of a manifold as defined in GP

I am reading Guillemin & Pollack's Differential Topology and I love it so far. However, the chapter about orientation of a manifold makes me struggle quite a bit. I have yet fully understood it. In particular, I am lost every time the book says…
Nick
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F is a submersion of $X \times S$, and for a fixed $x$.

The following text confuses me very much: (1) For fixed $x \in X$, is my understanding correct? $F_{(x,s)}$ takes $(x,s)$ and gives $f(x)+s$, so $dF_{(x,s)}$ takes $(x,s)$ and gives $df(x)+ds$. But $x$ is fixed, so $df(x) = 0$. Also, $ds$ is just…
1LiterTears
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How to draw the conclusion that $f$ is continuous?

Given $X$ is compact and $Y$ connected, and $f$ is a submersion. How to draw the conclusion that $f$ is continuous? In my book, submersion is defined as:
WishingFish
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How can I show that $a,b \in Z$?

I have a question to the end of my proof for the problem 1.3.10 on Guillemin and Pollack's Differential Topology: Generalizaition of the Inverse Function Theorem: Let $f: X \rightarrow Y$ be a smooth map that is injective on a compact sumbanifold…
WishingFish
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submanifold and open subset

If $U$ is an open subset of the manifold $X$, check that $$T_x(U) = T_x(X) \text{ for } x \in U.$$ I only proved when $U$ is an open subset of the manifold $X$, which is not true for submanifolds of $X$ in general right? My thought is…
1LiterTears
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