# 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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### Show that the central circle $X$ in the open Mobius band has mod 2 intersection number $I_2(X,X)=1$

Show that the central circle $X$ in the open Mobius band has mod 2 intersection number $I_2(X,X)=1$ Like in this picture http://i58.tinypic.com/2dkjwug.png Boundary Theorem: suppose that $X$ is ...
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### Show that if $X$ may be deformed into $Z$ then $X$ and $Z$ are cobordant.

Show that if $X$ may be deformed into $Z$ then $X$ and $Z$ are cobordant. Deformation definition: deformation of a submanifold $Z$ in $Y$ is a smooth homotopy $i_t:Z\to Y$ where $i_o$ is the ...
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### A question related to the topology of the level sets of a particular type of smooth functions $f:\mathbb{R}^2\to \mathbb{R}$.

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a smooth function without critical points; i.e. such that $\nabla f(x)\neq (0,0)$, for all $x\in\mathbb{R}^2$. Is it true or false that all the level curves of $f$ ...
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### Problem solving strategies in differential topology

I was wondering if there is a bag of tricks somewhere for differential topology and smooth manifold problems just like there is for analysis by prof. Tao http://terrytao.wordpress.com/2010/10/21/245a-...
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### Prove that $\deg_2 (f) \equiv q \mod 2$

Let $f:S^1→S^1$ be any smooth map. There exists a smooth map $g:\mathbb R \to \mathbb R$ such that $f(\cos(t),\sin(t) )=(\cos(g(t)),\sin(g(t) )$ and satisfying $g(2π)=g(0)+2πq$ for some integers $q$. ...
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### Prove that there exists a smooth map $g:R→R$ such that $f(cos(t),sin(t) )=(cos(g(t)),sin(g(t) )$ and satisfying $g(2π)=g(0)+2πq$ .

Let $f:S^1→S^1$ be any smooth map. Prove that there exists a smooth map $g:R→R$ such that $f(\cos(t),\sin(t) )=(\cos(g(t)),\sin(g(t) )$ and satisfying $g(2π)=g(0)+2πq$ . The book told me to show that ...
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### Prove that $N(Z;Y)$ is trivial if and only if there exists a set of $k$ independent functions $g_1,…,g_k$ for $Z$ on some set $U$ in $Y$.

Prove that $N(Z;Y)$ is trivial if and only if there exists a set of $k$ independent global defining functions $g_1,…,g_k$ for $Z$ on some set $U$ in $Y$. That is $Z=\{y∈U:g_1 (y)=0,…,g_k (y)=0\}$ ...
### Prove that there exists a diffeomorphism from an open neighborhood of $Z$ in $N(Z;Y)$ onto an open neighborhood of $Z$ in $Y$.
Prove that there exists a diffeomorphism from an open neighborhood of $Z$ in $N(Z;Y)$ (normal bundle of $Z$ in $Y$) onto an open neighborhood of $Z$ in $Y$. $\epsilon$ neighborhood theorem: For a ...
Given a $C^0$ function $g:[a,b]\to \mathbb{R}$ that is smooth everywhere except at $c$ (where $a<c<b$), and has positive derivative everywhere except at $c$, the claim is that there exists a ...