# Tagged Questions

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.

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### curl-free vector field on 3-torus

Let $U$ be an open and simply-connected subset of $\mathbb{R}^3$. Then for every curl-free vector field $v \: \colon U \to \mathbb{R}^3$ there is a potential $\phi \in C^{\infty}(U; \mathbb{R})$ such ...
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### Reference request for Thom's Transversality Theorem.

I am trying to read the book Introduction to the h-principle by Eliashberg and Mishachev. I am unable to understand the proof of Thom's Transversality theorem in the book. So if anyone can give any ...
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### Is true that a closed $k-$form on $T^n$, the torus, is exact if, and only if, the integral of $\omega$ over every compact submanifold is zero?

Is true that a closed $k-$form on $T^n$, the torus, is exact if, and only if, the integral of $\omega$ over every compact and oriented submanifold is zero?
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### How to prove that the Lefschetz number is invariante under homotopy?

How to prove that the Lefschetz number is invariante under homotopy? We define the Lefschetz number as the number of $f : M \to M$ as the number of intersection of the map $g(x) = (x,f(x))$ with the ...
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+100

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### Invariance of Linking numbers and critical values

So, I am trying to show that for a map $f: S^{2p-1} \rightarrow S^p$ , the linking number $l(f^{-1}(y),f^{-1}(z))$ of two framed submanifolds associated with regular values $y,z$ of $f$, defined as ...
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### Why every map $f : S^n \to T^n (n>1)$ has topological degree zero?

Why every map $f : S^n \to T^n (n>1)$ has topological degree zero? I don't know anything about covering spaces, and has been told to me that this assertion comes from this theory! I do appreciate ...
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### Let $\gamma_1, \gamma_2 : I \to M$ two homotopic curves, $M$ a $C^{\infty}$ manifold. If $\omega$ is a closed $1-$form then:

Let $\gamma_1, \gamma_2 : I \to M$ two homotopic curves, $M$ a $C^{\infty}$ manifold. If $\omega$ is a closed $1-$form then: $$\int_{\gamma_1(I)} \omega = \int_{\gamma_2(I)}\omega.$$ Then, conclude ...
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### Area of a domain with Stokes' Theorem

This question came up on a preliminary exam: Define $$g(s,t)=(x(s,t),y(s,t))=(\cos(s)+\cos(t),\sin(s)+\sin(t)),$$ on the region $-\pi<s<\pi$, $s<t<s+\pi$. (The function $g$ is one-...
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### What does it take for a smooth homeomorphism to be a diffeomorphism?

I have an open subset $A$ of $\mathbb{R}^k$ and a subset $B$ of $\mathbb{R}^n$, $n>k$, that are homeomorphic and $f:A\longrightarrow B$ is a smooth homeomorphism between two sets. I'm wondering if ...
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### Let $\deg$ be the topological degree. Then $\deg(fg) = \deg(f)\deg(g)$, with $f, g : M \to N$

Recall that the topological degree is defined as: Let $f : M \to N$ a $C^k$ function and $y$ be a regular value of $f$. Then we define: $$\deg(f)= \sum_{f(x) = y}|Df(x)|,$$ where $| . |$ means the ...
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### Understanding Milnor's proof of the fact that the preimage of a regular value is a manifold

In the book "Topology from the Differential Viewpoint" (Milnor) he proves on page 11 the following lemma: If $f: M\to N$ is a smooth map between manifolds of dimension $m\geq n$ and if $y\in N$ is ...
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### Is the Inverse of a Coordinate System on a Differentiable Manifold Differentiable?

In his book Calculus on Manifolds, Spivak shows that a differentiable k-dimensional manifold in $\mathbb{R}^n$ is a set $M$ such that $\forall x \in M$ there exists an open set $U$ containing $x$, an ...
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### non-homotopic maps $\mathbb S^2\rightarrow \mathbb{P}_\mathbb{R}^2$

How can I find non-homotopic maps $\mathbb S^2 \rightarrow \mathbb{P}_\mathbb{R}^2$? I know that it is enought that the degree are different. But the canonical map given by the antipodal map has ...
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### Applications of Banach's fixed point theorem on Differential Geometry

Does anyone know any simple application of Banach's fixed point theorem on Differential Geometry. I am looking for something involving manifolds.f
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### Transition maps on Grassmanian $Gr(2,5)$

I need to provide charts and transition maps on Grassmanian (2, 5). (All 2-dimensional subspaces in 5-dimensional space). I know how the charts look, used definition from this document: http://people....
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### Can I conclude $s$ is a submersion from these data?

Let $M$ and $N$ be smooth manifolds ($C^\infty$). Let $s\in C^\infty(M, N)$ and $u\in C^\infty(N, M)$ be maps satisfying: $u$ is an embedding; $s\circ u=\textrm{id}_N$; $(u\circ s)^2=u\circ s$. ...
### Proving that an $E$-oriented manifold has an $E$-oriented normal bundle
This is the setting we are working in: $M$ is a closed, smooth $n$-manifold embedded in $\mathbb{R}^{n+k}$ with a chosen embedding $e\colon M^n\to \mathbb{R}^{n+k}$. It is $E$-oriented, for $E$ a ...
What is the topological degree of the constant map? To me it does not make any sense, once $f$ being the constant map has no regular values. So, how to proceed?