# 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.

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### Reference request: Operation on Regular Homotopy Classes

Recently, I stumbled upon the definition of regular homotopy classes and Smale's Theorem as in Smale, S. (1958). Regular curves on Riemannian manifolds. Transactions of the American Mathematical ...
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### Degeneration of nodes and punctures on Riemann surfaces

It is well known that it is possible to build Riemann surfaces with singularities such as nodes (like a pinched torus) or with punctures. Now a node on a torus can be obtained by shrinking a circular ...
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### Proving $S^{1}$ is a manifold under the quotient topology

Lets say I have defined $S^{1}$ as $[0,1]/\sim$ where $x\sim y$ when $x=y$ except when $x,y$ equal 0 or 1. In other words, $[0,1]/\sim$ is the singletons and the set $\{0,1\}$. I want to show that ...
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### Constructing unique $2:1$ covering of a non-orientable, connected, smooth manifold

Let $M$ be a non-orientable, connected, smooth manifold with $\text{dim }M=n$. I'm trying to fill in the ideas of a construction of a unique $2:1$ covering $\tilde{M}$ of $M$. I've pieced together ...
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### Suppose for a region on a surface I can draw a "handle" can I cut the surface to reduce it's genus while leaving the region intact?

Suppose I have a smooth orientable surface $Q$ and a compact region $R$ of $Q$. Suppose there is a closed curve $C$ that divides R into two connected components $R_1,R_2$ but does not divide Q into ...
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### How to determine the fundamental group of $\{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\}$.

Consider a manifold $N$ defined as follows $$N=\{n\otimes n-m\otimes m:n,m\in S^2,\,\,(n,m)=0\}\subset M^{3\times 3},$$ where $S^2$ denotes the two dimensional sphere, $(\cdot,\cdot)$ ...
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### The torus is in $\mathbb{R}^{3}$ or $\mathbb{R}^{4}$?
Always I belived that the torus is in $\mathbb{R}^{3}$, but reading about the clifford Tori I read that the Clifford torus is in $\mathbb{R}^{4}$. I don't understand it. Since in differential ...