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Results tagged with differential-topology
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user 1450
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”.
4
votes
Which continuous functions are polynomials?
I'm going to go out on a limb and make a partial conjecture based on Tom Goodwillie's comment. A function $f:\mathbb{R}^n \to \mathbb{R}$ is topologically conjugate to a polynomial $p(x)$ only if it …
- 56.6k
2
votes
Does exist a $\varepsilon$-tubular neighborhood of a smooth complex quasi-affine algebraic v...
If I understand the question correctly, no because $V \subset \mathbb{C}^2$ could be the union of $xy = 1$ with $x=0$. You can see from looking at the real solutions that it does not have a tubular n …
- 56.6k
23
votes
Accepted
Smooth bijection between non-diffeomorphic smooth manifolds?
Every smooth manifold has a smooth triangulation, which yields a pseudofunctor from the category of smooth manifolds to the category of PL manifolds. (There is no actual functor; that would be crazy. …
- 56.6k
11
votes
Accepted
Morse Theory and Exotic Spheres
Milnor didn't explain the formula as much as maybe he should have, but the point is that the real part of a unit-length quaternion is invariant under both conjugation and inversion. Let $$r = ||u|| \ …
- 56.6k
47
votes
Can every manifold be given an analytic structure?
There is an amazing theorem of Morrey and Grauert that says that not only does every (paracompact) smooth manifold have a real analytic structure, the real analytic structure is unique. Using Whitney …
- 56.6k
38
votes
Accepted
Exotic differentiable structures on R^4?
I once heard Witten say that topology in 5 and higher dimensions "linearizes". What he meant by that is that the geometric topology of manifolds reduces to algebraic topology. Beginning with the Whi …
- 56.6k
5
votes
Can homologous submanifolds be connected by an immersed manifold with boundary?
First of all, it doesn't matter whether or not the map is smooth. If you find any continuous map, then it will have a smooth approximation.
The other answers so far explain that the cobordism group …
- 56.6k
17
votes
Accepted
Embeddings of $S^2$ in $\mathbb{CP}^2$
The conjecture that every $S^2 \subseteq \mathbb{C}P^2$ is standard if it is homologous to flat is implied by the smooth Poincaré conjecture in 4 dimensions. It also implies a special of smooth Poinc …
- 56.6k