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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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Action of U(1) on a sphere bundle

Suppose $N$ is a closed, $n$-dimensional, orientable smooth manifold. Moreover, $n$ is odd. Consider the tangent bundle $TN$ of $N$. By adding a point at infinity to each tangent space I define a sphere bundle $E$ over $N$. Notice, that every…
Piotr Pstrągowski
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Proving a Certain Smooth Map $S^n\rightarrow S^n$ is a Diffeommorphism

I am given a smooth map $f:S^n\rightarrow S^n$, for $n\geq 2$, whose differential is injective at each point. I am asked to prove that it is a diffeomorphism. Since the differential is injective between manifolds of the same dimension, it is also…
J126
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Why is every derivation a vector?

We can see the vectors of the tangent space $T_pM$ to a smooth manifold as velocities of curves. This is elaborated here. Each velocity $\gamma'(0)$ corresponds to a derivation $D_{\gamma}(f) = (f \circ \gamma)'(0)$ as seen in the Wikipedia…
Jean Valjean
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proving jordan brouwer separation theorem

For proving Jordan separation theorem in differential manifold theory, one step involves proving the following: Let $z\in{\mathbb{R}^n}\setminus X$, where $X$ is connected, closed manifold of dimensin $n-1.$ suppose that $x\in{X}$ and $U$ is an…
user1123
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the second projection is a local diffeomorphism between manifods with boundary

Good afternoon, I Have problems with these exercises, I wrote the two questions because the second one needs the result of the previus one. They are: Let $n,m, k=n+m$ positive integers, so we can identify $\mathbb R^k\cong\mathbb R^n\times \mathbb…
Framate
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What exactly are the basis $\{ \frac{\partial}{\partial x_i}\mid_p \}$ of the tangent space of a manifold?

From http://en.wikipedia.org/wiki/Tangent_space#Definition_via_derivations, I understand that if $\gamma: (-1,1) \to M$ is a curve (and $M$ a manifold), with tangent vector $\gamma'(0)$, then the corresponding derivation is $D_{\gamma}(f) =…
Jean Valjean
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Is this a local diffeomorphism?

I want to find a local diffeomorphism $\Bbb{R}^2\to\Bbb{R}^2$ that is not a diffeomorphism onto its image. This is what I thought: $f(x,y)=(\sin 2\pi x, \cos 2\pi y)$. Does that work? Seems ok to me.
helix
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Differentiable function on bad sets.

This is exercise (c) on page 6 of ``Elementary differential topology" by Munkres. Find an open subset $U$ of $\mathbb R^2$ and a $C^1$ map $f : A \to \mathbb R$ ($A = \overline U$) such that $Df(x)$ ($x \in A$) depends on the extension of $f$ to a…
Compact Support
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Determine $n$ so that manifold is locally homeomorphic to $\mathbb{R}^n$?

I just started studying smooth manifolds. The definition of a topological manifold requires a topological space to be locally Euclidean: homeomorphic to $\mathbb{R}^n$. I know some examples, like how a 2-sphere is locally homeomorphic to…
Jean Valjean
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Intersection number on $S^k$

Is the intersection number on $S^k$ is always zero? Choose $X$ a compact submanifold of $S^k$, and $Z$ a closed submanifold of complementary dimension, viewing $I_2(X,Z) = I_2(i, Z)$ where $i: X \hookrightarrow Y$ is the inclusion. Then $X,Z$ can…
1LiterTears
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Confused on Guillemin and Pollack's proof of the $\epsilon$-Neighborhood theorem.

On pg. 71 of Guillemin and Pollack they prove the $\epsilon$-Neighborhood theorem. Here $Y$ is a compact boundaryless manifold in $\mathbb{R}^M$. They say Proof: Let $h:N(Y)\to\mathbb{R}^M$ be $h(y,v)=y+v$. Notice that $h$ is regular at every point…
Geovanna Anthony
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How to prove there exists a solution? Guillemin Pollack

Prove there exists a complex number $z$ such that $$ z^7+\cos(|z^2|)(1+93z^4)=0. $$ (For heaven's sake don't try to compute it!)
jimbo
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Every chart is smoothly compatible, what is wrong with my argument?

Let $A_1$ and $A_2$ be two distinct smooth structures on the manifold $M$. For any two charts $(U,\phi)\subset A_1$ and $(V,\psi)\subset A_2$, both $\phi$ and $\psi$ are diffeomorphisms. If we assume that $U$ and $V$ are not disjoint, then…
gedeon
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Extending an embedding from a compact submanifold

Suppose $X, Y$ are smooth manifolds, $Z \subset X$ a compact submanifold and $f: X \rightarrow Y$ a smooth map such that the restriction of $f$ to $Z$ is an injective smooth immersion. As $Z$ is compact, the restriction is actually a smooth…
Paul
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Degrees are the only interesting intersection numbers on spheres

Show that if $f: X \rightarrow S^k$ is smooth, $X$ compact and $0 < dim(X) < k$, then for all closed $Z \subseteq S^k$ of dimension complementary to $X$, we have $I_2(X, Z) = 0$. An idea：let p be a regular value of f, using Sard,which is to say a…
user105536
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