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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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Show that $f(C)$ has Hausdorff dimension at most zero.

We say that a set $A \subset \mathbb{R}^n$ has $d$-dimensional Hausdorff measure zero if for all $\epsilon > 0$ there exists a covering of $A$ by countably many cubes $S_i$ with side lengths $s_i$ such that $\sum_i (s_i)^d < \epsilon$. The Hausdorff…
1LiterTears
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Showing a diffeomorphism extends to the neighborhood of a submanifold

Does anyone have a proof of problem 14, on page 56 of Guillemin and Pollack? I meant to do it as an exercise (I'm teaching myself the subject) but I'm struggling with the last step. Suggestions? Suppose that the derivative of $f: X \rightarrow Y$ is…
Eric
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Problem from "Differential topology" by Guillemin

I am strugling one of the problems of "Differential Topology" by Guillemin: Suppose that $Z$ is an $l$-dimensional submanifold of $X$ and that $z\in Z$. Show that there exsists a local coordinate system $\left \{ x_{1},...,x_{k} \right \}$ defined…
Tomas
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Do there exist exotic tangent bundles?

One knows that the tangent bundle of an exotic sphere is bundle isomorphic to the tangent bundle of the standard sphere. Are there closed manifolds that admit several different differentiable structures in which two of the tangent bundles are not…
lavinia
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is the fixed set of a smooth involution a submanifold?

Let $f:X\rightarrow X$ be a smooth map of a smooth manifold with $f^2=\operatorname{id}$. Is the subset $\{x\in X\mid f(x)=x\}$ a smooth submanifold? I tried to find an argument with the implicit function theorem, but I don't have an answer.
fer
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Retracts are Submanifolds

Looking over some old qualifying exams, we found this: Let $A\subseteq M$ be a connected subset of a manifold $M$. If there exists a smooth retraction $r:M\longrightarrow A$, then $A$ is a submanifold. Our thought to prove this statement was that…
J126
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Diffeomorphisms and Stokes' theorem

Problem: Let $\omega\in\Omega^r(M^n)$ suppose that $\int_\sum \omega = 0$ for every oriented smooth manifold $\sum \subseteq M^n$ that is diffeomorphic to $S^r$. Show that $d\omega = 0$. Proof: Assume $d\omega \neq 0$. Then there exists $v_1,…
M.B.
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Proper maps and families of compact complex manifolds

Kodaira defines a complex analytic family of compact complex manifolds as the data $(E,B,\pi)$, where $E$ and $B$ are complex manifolds, and $\pi$ is a surjective holomorphic submersion such that the preimage $\pi^{-1}(x)$ of any point $x \in B$ is…
Brendan Fong
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Conceptual error in Kosinski's "Differential Manifolds"?

This seems like such an egregious error in such an otherwise solid book that I felt I should ask if anyone has noticed to be sure I'm not misunderstanding something basic. In his section on connect sums, Kosinski does not seem to acknowledge that,…
James
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Why is there no foliations of the 2-sphere, or a genus two surface?

I'm trying to see why there is no (one-dimensional) foliation of $S^2$ or an orientable surface of genus two. Originally I was thinking that such a foliation could give me a non-vanishing vector field, which would be a contradiction, but now I have…
paragon
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Reference for self-intersections of immersions

I believe the following is a true fact: if $f : S^1 \to \mathbb{R}^2$ is an immersion, there is an arbitrarily small homotopy of f to another immersion $g : S^1 \to \mathbb{R}^2$ such that $g$ only intersects itself finitely many times. This…
Pedro
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Local Submersion Theorem - Differential Topology of Guillemin and Pollack

Local Submersion Theorem: Suppose that $f:X \to Y$ is a submersion at $x$, and $y=f(x)$. Then there exist local coordinates around $x$ and $y$ such that $f(x_1,\dotsc,x_k)=(x_1,\dotsc,x_{\ell})$. That is, $f$ is locally equivalent to the canonical…
user230283
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Prove that if $S^k$ has a non vanishing vector field, then its antipodal map is homotopy to the identity if $k$ is even.

Prove that if $S^k$ has a non vanishing vector field, then its antipodal map is homotopy to the identity if $k$ is even. Here is what I got so far. Suppose we have an antipodal map $x \to -x$ of $S^k \to S^k$. For $k=1$ the antipodal just rotating…
Diane Vanderwaif
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A question regarding the proof of Hopf's theorem.

This is a question regarding the proof of the Hopf theorem given in "Topology from a Differential Viewpoint" by Milnor: If $v:X\to \Bbb{R}^m$ is a smooth vector field with isolated zeroes, and if $v$ points out of $X$ along the boundary, then the…
user67803
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Atlas on a smooth manifold that contains 2 charts

How can I show that if an atlas on a smooth manifold has exactly 2 charts then it is orientable? How do I make sure that the Jacobian of the transition map is positive?
bluebox
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