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Results tagged with differential-topology
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user 121
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”.
7
votes
Accepted
Triangulations of exotic 4-spheres
Here is my comment expanded to answer form: The question of existence of exotic 4-spheres (i.e., the smooth Poincaré conjecture) is still open, and (according to Wikipedia) the existence of exotic PL …
- 45.7k
5
votes
compact riemann surface of genus g
Choose $2g+2$ distinct complex numbers $z_i$, and take a double cover of $\mathbb{P}^1$ branched at these points. This is typically written as a plane curve with an affine patch defined by $y^2 = \pr …
- 45.7k
5
votes
Are there good product rules on the $k$-sphere?
One condition we might like for a product on a manifold to be nice is that it admits a 2-sided identity element. Another condition we might like is that left multiplication is nonsingular near the id …
- 45.7k
6
votes
Can we decompose Diff(MxN)?
I'm not an expert, but my impression was that you can't reasonably expect anything like a decomposition in general. Here is a big list of references on automorphisms of manifolds, compiled by Andre H …
- 45.7k
20
votes
Theoretical physics: Why not just $\mathbb{R}^4$?
I can answer your literal question. Not everyone studies exotic $\mathbb{R}^4$, because the universe of mathematical and theoretical physics is a big one with many interesting ideas, and there's no r …
- 45.7k
5
votes
Why is the dual of a torus the same as its fundamental group?
This is just a minor elaboration on David Ben-Zvi's answer. You can see the duality between the fundamental group of $T$ and the character lattice by composing based loops $\mathbb{R}/\mathbb{Z} \to …
- 45.7k