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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.

7287 questions
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Showing that a level set is not a submanifold

Is there a criterion to show that a level set of some map is not an (embedded) submanifold? In particular, an exercise in Lee's smooth manifolds book asks to show that the sets defined by $x^3 - y^2 = 0$ and $x^2 - y^2 = 0$ are not embedded…
JohnSharp
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When do regular values form an open set?

Let $f:M\to N$ be a $C^\infty$ map between manifolds. When is the set of regular values of $N$ an open set in $N$? There is a case which I sort of figured out: If $\operatorname{dim} M = \operatorname{dim} N$ and $M$ is compact, it is open by the…
Bruno Stonek
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Fixed point theorem on spheres

In Milnor's book Topology from the Differentiable Viewpoint there's the following problem: Problem $6$ (Brouwer). Show that any map $S^n\to S^n$ with degree different from $(-1)^{n+1}$ must have a fixed point. My solution: Assume that the map…
Daniel Robert-Nicoud
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Tangent bundle of $S^1$ is diffeomorphic to the cylinder $S^1\times\Bbb{R}$

How do I construct an explicit diffeomorphism between $TS^1$ and $S^1\times\Bbb{R}$? It will be something like $\phi:TS^1\to S^1\times\Bbb{R}, (x,v)\to(x,...)$. Also we know that for $x=(x_1,x_2)$ and $v=(v_1,v_2)$, $x_1^2+x_2^2=1$ and…
bluebox
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Folliation and non-vanishing vector field.

The canonical foliation on $\mathbb{R}^k$ is its decomposition into parallel sheets $\{t\} \times \mathbb{R}^{k-1}$ (as oriented submanifolds). In general, a foliation $\mathcal{F}$ on a compact, oriented manifold $X$ is a decomposition into $1-1$…
WishingFish
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Can we smoothly embed $\mathbb{S}^2 \times \mathbb{S}^1$ or $\mathbb{RP}^2 \times \mathbb{R}$ in $\mathbb{R}^4$?

I've been thinking a little bit about smooth embeddings recently. In particular, I was wondering: Do the $3$-manifolds $\mathbb{S}^2 \times \mathbb{S}^1$ and $\mathbb{RP}^2 \times \mathbb{R}$ embed smoothly into $\mathbb{R}^4$? As with most of…
Jesse Madnick
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Does having a codimension-1 embedding of a closed manifold $M^n \subset \mathbb{R}^{n+1}$ require $M$ to be orientable?

I'm trying to follow a proof about immersing/embedding $\mathbb{RP}^n$ into $\mathbb{R}^{n+1}$, which goes roughly as follows: Write $\tau=T\mathbb{RP}^n$. The normal bundle $\nu$ has rank 1, so its Steifel-Whitney class is $w(\nu)=1$ or…
Aaron Mazel-Gee
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Show every $f_t$ is Morse for $t$ is sufficiently small

Let $f$ be a Morse function on the compact manifold $X$. Let $f_t$ is a homotopic family function with $f_0=f$. Show every $f_t$ is Morse for $t$ is sufficiently small Here is my argument, but my professor said it's not correct, without telling me…
XiaoXiao Zhen
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Is there a characteristic property of quotient maps for smooth maps?

If $\pi\colon X\to Y$ is a quotient map, and $f\colon Y\to Z$ is a continuous map between topological spaces, then the characteristic property of the quotient map says $f$ is continuous iff $f\circ \pi$ is continuous. Does this also work for smooth…
Charli
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Question about image of proper smooth map of constant rank. (Undergraduate)

I have to find a proof of the following theorem Given a smooth function f between smooth manifolds X and Y that: has constant rank is proper the preimage of every point in f(X) is connected and simply connected Then f(X) is a smooth submanifold of…
Bankovir
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normal bundle of level set

Let $M$ be a Riemannian manifold and $S \subset M$ a regular level set of a smooth function $f:M\rightarrow \mathbb{R}^k$. How can I show that the normal bundle of $S$ is trivial? If $k=1$ then $\text{grad}f$ is a global frame for $NS$ but I am not…
Manuel
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Compute $\chi(\mathbb{C}\mathrm{P}^2)$.

I need to compute $\chi(\mathbb{C}\mathrm{P}^2)$ using techniques from differential topology. I cannot think of any theorems that are particularly useful for this computation, so I think that I will have to find a vector field on…
jgensler
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Embedding, local diffeomorphism, and local immersion theorem.

Suppose $f: M \to N$ is smooth and an immersion, i.e $df_p : T_p(M) \to T_p(N)$ is one-to-one. Since $f$ is an immersion, we have the following theorem, $\textbf{Local Immersion Theorem:}$ Suppose that $f: M \to N$ is an immersion at $x$. Let…
Yuugi
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Showing Degree $1$ Maps Induce Surjections on $\pi_1$

I am running a qualifying exam prep course. A question I posed to my students was: Suppose that $M$ and $N$ are compact, oriented manifolds and $f:M\longrightarrow N$ is of degree $1$. Show that $f_*:\pi_1(M)\longrightarrow \pi_1(N)$ is a…
J126
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A Surjective Local Smooth Diffeomorphism That is Not A Covering Map

Let $\pi:M_1\rightarrow M_2$ be a surjective $C^{\infty}$ map between two connected manifolds with $d\pi$ an isomorphism. If $M_1$ is compact, it is seen that $|\pi^{-1}(m_2)|$ is finite, so $\pi$ is a covering. If we only have $M_2$ compact, do we…
Anonymous Coward
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