# 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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### 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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### 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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### 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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### 2-connected 6 manifolds with boundary $S^5$

What are the 2-connected 6-manifolds that have boundary $S^5$? Are they all of the form $(\sharp_{i=1}^k S^3 \times S^3) \backslash D^6$ for some $k \ge 1$? Also, I think if $M^5$ is simply-...
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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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### 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 $x=(x_1,x_2)$ ...
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### Hodge theory on semi-Riemannian manifolds [reference request]

I need to learn a bit of Hodge theory on manifolds and I am looking for a reference which covers the case where the metric has arbitrary signature $(p,q)$. Most books I have found seem to focus on the ...
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