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Results tagged with differential-topology
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user 5010
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
14
votes
1
answer
1k
views
Who proved that two homotopic embeddings of one surface in another are isotopic?
If $\Sigma_1$ and $\Sigma_2$ are two compact topological surfaces with boundary and $\phi, \psi : \Sigma_1 \hookrightarrow \Sigma_2$ are two orientation-preserving embeddings that are homotopic, then …
- 683
5
votes
Decomposition of a closed surface
It's the same proof. Take a topological pants decomposition as before, and look for a minimal-length representative on your given Riemannian metric. Then you invoke the theorem that if you have a simp …
- 10.1k
3
votes
Accepted
Computing the hopf invariant (without integration or homology, as in Milnor) of the hopf map
If you have the Hopf link embedded in some standard way in $\mathbb{R}^3$, you can see the linking number as given by the degree of a map $S^1 \times S^1 \to S^2$ in a number of ways. For instance, t …
- 10.1k