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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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manifolds and curved spaces

The book on general relativity I'm reading states that the motivation behind the concept of manifold is to extend the theory of analysis in $ℝ^n$ to curved spaces. As an example of manifold, it gives the 2-sphere $S^2$ \begin{equation} S^2 = \{x,y,z…
Michaelangelo Meucci
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On the immersion, regular value theorem

Let $g\colon N\to M$ be an immersion. Then, I think that $g^{-1}(p)$ is finite set or $0$-dimensional manifolds for all $p\in M$. Now, let $g_t\colon N\to M$ be an one-parameter family of an immersion. Let $F\colon N\times [0,1]\to M$ be the…
user9236
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A Lemma from Milnor's Topology from the Differentiable Viewpoint

I have a question about an proof in Milnor's TOPOLOGY FROM THE DIFFERENTIABLE VIEWPOINT (see pages 65-66): The aim is to show the classifying theorem that any smooth, connected $1$-dimensional manifold is difeomorphic either to the circle $S^1$ or…
user267839
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True or False in differential topology

I am trying to answer the following questions with a True (and give a proof) or a False (and give a counter example). I really have no idea how to approach this problems or how to start thinking about them. I would appreciate any hints, comments or…
BOlivianoperuano84
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Is the image of $f: R \to S^1 \times S^1, f(t) = (e^{it}, e^{\sqrt 2it})$ a submanifold?

Let $f: R \to S^1 \times S^1$ be $f(t) = (e^{it}, e^{\sqrt 2it})$,is $f(\mathbb R)$ a submanifold? By easy verification, $f$ is an injective immersion. Then if I want to show that the image is a submanifold, I will want to show that $f$ is a smooth…
penny
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Proving a set is not a embedded submaifold.

So I am asked to find out which level sets of the function $f:\mathbb{R}^2 \to \mathbb{R}$ given by $f(x,y)=x^3+xy+y^3+1$ are embedded submanifolds of $\mathbb{R}^2$. You can see that the points in the set $X=\mathbb{R}-\{1,\frac{28}{27}\}$ are…
Bajo Fondo
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Example of transversal map f where boundary of preimage is not preimage intersect boundary

I am utterly confused by this example in Guillemin and Pollack gave in "differential topology" page 60. For a manifold with boundary, We would like conditions that would guarantee that if $f: X \rightarrow Y$ encounters a submanifold Z of Y, then…
Ecotistician
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Guillemin and Pollack proof of Borsuk-Ulam

This question is unfortunately a 3-in-1 question, because Guillemin and Pollack's proof of Borsuk-Ulam relies on exercise 2.6.1 and exercise 2.6.2, the latter of which relies on exercise 2.4.8. I actually think I can do 2.6.1 and 2.6.2, but 2.4.8 is…
Tanner Strunk
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If $M$ is a $k$-dimensional manifold, show that every point of $M$ has a neighborhood homeomorphic to all of $\mathbb{R}^k$ .

If $M$ is a $k$-dimensional manifold, show that every point of $M$ has a neighborhood homeomorphic to all of $\mathbb{R}^k$ . Therefore, charts can always be chosen with all of Euclidean space as their co-domains. I'm confused because I thought that…
user394412
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Show that 1 is a regular value of f. Identify the manifold $M=f^{-1}(1)$.

Consider the real valued function $f(x,y,z) = (2-(x^2+y^2)^{\frac{1}{2}})^2 + z^2$ on $R^3-{(0,0,z)}$. Show that 1 is a regular value of f. Identify the manifold $M=f^{-1}(1)$. I showed that 1 is regular value of f by proving that $f_*$ is onto for…
Yeezus
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determining whether a function from the torus to R4 is an embedding

I am not sure whether the following question requires me to find explicit charts or whether there is more theoretical machinery that may be used. Starting from the description of $S_1$ as the unit circle in $\mathbb{R}^2$, we can identify the…
TerryL
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$\alpha \wedge d\alpha=0$ on 3 manifold for a $1$-form $\alpha$

Is it true that, for any $1$-form on a $3$-mfld $\alpha$, $\alpha \wedge d\alpha=0$? I'd argue like $$0=d(0)=d(\alpha \wedge \alpha)= 2d(\alpha)\wedge \alpha$$ since for any $1$-form $\alpha \wedge \alpha=0$, but then there are counterexample like…
Luigi M
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Milnor's proof of the Homotopy Lemma

I'm reading through Milnor's Topology from the Differentiable Viewpoint, and one page $21$ he does the following Homotopy Lemma: Let $f, g : M \to N$ be smoothly homotopic maps between manifolds of the same dimension, where $M$ is compact and…
Perturbative
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Find a Line Not Tangent to Any Point on a Curve

I am given the following: Let $C$ be a simple closed curve in $\mathbb{R}^2$ and let $p \in C$. Show that there exists a line in $\mathbb{R}^2$, passing through $p$, that is not tangent to $C$ at any point. First, we can assume that $p$ is the…
J126
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$\Omega^*$ and $T^*$

Let $\Omega^*$ be the algebra generated by $dx_1,\dots,dx_n$ subject to the relations $(dx_i)^2=0$ and antisymmetry. Does this form the cotangent space $T^*$? I thought it might, since the tangent space $T$ has basis $\left\{\frac{\partial}{\partial…
Earth Cracks
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