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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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An exercise in Spivak's *Calculus on Manifolds*

Problem 5-6 in Michael Spivak's Calculus on Manifolds reads: If $f:\mathbb R^n\to\mathbb R^m$, the graph of $f$ is $\{(x,y):y=f(x)\}$. Show that the graph of $f$ is an $n$-dimensional manifold if and only if $f$ is differentiable. (here manifold…
Mariano Suárez-Álvarez
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A submersion $F : \mathcal{X} \to \mathcal{Y}$ must be surjective.

Let $\mathcal{X}$ and $\mathcal{Y}$ be compact manifolds and let $\mathcal{Y}$ be connected. Prove that a submersion $F : \mathcal{X} \to \mathcal{Y}$ must be surjective. I don't have much thought on this question, except for if $F$ is not…
1LiterTears
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Showing that the exponential map $\mathrm{exp}:\mathfrak{sl}(2,\mathbb{R})\to\mathrm{SL}(2,\mathbb{R})$ is not surjective

I am having a difficult time showing that the exponential map $\mathrm{exp}: \mathfrak{sl}(2, \mathbb{R}) \rightarrow \mathrm{SL}(2, \mathbb{R})$ is not surjective. I have, however, worked out that $\mathfrak{sl}(2, \mathbb{R})$ is given by $\{A…
Susan
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Typos in Getzler's "Short proof of local Atiyah-Singer index theorem"?

I am a physicist reading Ezra Getzler's paper A short proof of the local Atiyah-Singer index theorem (Topology 25 (1986) 111-117). The clever step in his proof is his rescaling so as to focus of the short distance behaviour of the heat kernel for…
mike stone
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Natural projection of tangent bundle is submersion

I am wondering how to see that the natural projection from $\pi:TX \to X$ is a submersion. To see that $\pi$ is surjective I can identify $TX$ as product space $X \times F$ where $F$ is a vector space with the same dimension as $X$ and so for all…
JDoe
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Hairy dog theorem

I am interested in applications of the hairy ball theorem to non-closed surfaces, such as dogs and cats. As has been pointed out to me in my previous question, it is possible to arrange the two cowlicks of a hairy ball so that they are on top of…
Superbest
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Mapping Degree of a Smooth Map from a Compact Manifold without Boundary

There is a comment in Milnor's "Topology from a Differentiable Viewpoint," that I don't quite understand: Let $f$ be a smooth map from $M$ to $N$, where $M$ is compact without boundary, and $N$ is connected, and both manifolds have the same…
Braindead
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Show that every k-dimensional vector subspace V of $R^N$ is a manifold diffeomorphic to $R^k$.

I'm actually in a exercise of the book " Differential Topology " of Guillemin and Pollack. Show that every k-dimensional vector subspace $V$ of $R^N$ is a manifold diffeomorphic to $R^k$, and that all linear maps on $V$ are smooth. Definition : …
user230283
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Diffeomorphism from level set onto $S^2$

I'm given a map $\phi : R^4 \rightarrow R^2$ defined by $\phi(x,y,s,t) = (x^2 + y, x^2+y^2+s^2+t^2+y)$. It's easy to show that the level set $C = \phi^{-1}(0,1)$ is a smooth submanifold of $R^4$ with dimension $2$ by showing that $(0,1)$ is a…
Saigyouji
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Show that the central circle $X$ in the open Mobius band has mod 2 intersection number $I_2(X,X)=1$

Show that the central circle $X$ in the open Mobius band has mod 2 intersection number $I_2(X,X)=1$ Like in this picture http://i58.tinypic.com/2dkjwug.png Boundary Theorem: suppose that $X$ is the boundary of some manifold $W$ and $g: X \to Y$ is…
XiaoXiao Zhen
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Prove that there exists a smooth map $g\colon R\to R$ such that $f(\cos(t),\sin(t))=(\cos(g(t)),\sin(g(t)))$ and satisfying $g(2π)=g(0)+2\pi q$.

Let $f\colon S^1\to S^1$ be any smooth map. Prove that there exists a smooth map $g\colon\mathbb{R}\to\mathbb{R}$ such that $f(\cos(t),\sin(t))=(\cos(g(t)),\sin(g(t)))$ and satisfying $g(2π)=g(0)+2\pi q$. The book told me to show that…
Diane Vanderwaif
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Is there such kind of theorem saying two homotopic ways of attaching handle result in diffeomorphic manifolds?

M' is a manifold with boundary. One can attaching a handle $h:=D^k\times D^{n-k}$ along $f:S^{k-1}\times D^{n-k}\rightarrow M'$ forming $M=M'\cup_f h$. Suppose f is homotopic with f', and that f,f' are smooth embeddings. Is there a theorem…
Honglu
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Do exotic smooth structures on ${\mathbb R}^4$ vary smoothly?

Is the two-parameter family of exotic smooth structures on ${\mathbb R}^4$ constructed by Taubes and Gompf a smooth family? In other words, is there a submersion $p:V\to\mathbb{R}_+\times\mathbb{R}_+$ of $C^\infty$-manifolds such that…
Dima Roytenberg
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Definition of pullback.

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: $$f^*\omega(x) = (df_x)^*\omega[f(x)].$$ Why it uses a…
WishingFish
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Help with Kervaire paper

I was trying to read Kervaire's 1960 paper where he first shows the existence of a manifold that does not admit differentiable structure and I got stuck. On the second page of the paper where he starts to define his invariant he writes: where $u_2…
snailspace
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