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Results tagged with differential-topology
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user 18050
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”.
48
votes
Example of a manifold which is not a homogeneous space of any Lie group
Apart from already mentioned non simply connected examples most simply connected manifolds are also not homogeneous. One easy criterion is that simply connected homogeneous spaces are rationally elli …
- 7,842
22
votes
Proving the existence of good covers
you don't really need a whole lot of Riemannian geometry to prove this. Embed the manifold into $\mathbb R^n$ by Whitney and look at very small charts around points given by orthogonal projections ont …
- 7,842
20
votes
Accepted
A symmetric embedding of manifolds
No. There exist closed manifolds which do not admit any compact group actions, in particular, no $Z_2$ actions. For example, Shultz showed in "Group actions on hypertoral manifolds. II." that in dime …
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16
votes
Accepted
Consequences of Gromov's Conjecture
As Igor mentioned knowing the optimal bound is always better than knowing a non-optimal one such as the bound provided by Gromov's proof. It rules out a lot more examples. A proof of the sharp bound w …
- 7,842
14
votes
Accepted
Metric deformations from non-negative to positive curvature
As Benoît Kloeckner points out this is false for non simply connected manifolds with $RP^2\times RP^2$ being a counterexample (by Synge's theorem). For simply connected manifolds this is a well known …
- 7,842
13
votes
Smooth representatives for elements of $\pi_7(\text{exotic $S^7$})$
Durán wrote down an explicit formula for such map in "Pointed Wiedersehen Metrics on Exotic Spheres and Diffeomorphisms of $S^6$". That is he wrote an explicit formula for an exotic diffeomorphism fr …
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12
votes
When is there a submersion from a sphere into a sphere?
I want to add that having a fiber bundle with total space a sphere is very restrictive even if you don't assume that the base is a sphere too. This has been studied. I will exclude the obvious cases o …
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11
votes
Are Hölder manifolds a thing?
I don't think this has been studied but I would note two things.
First, Sullivan proved that in dimensions $n\ne 4$ any topological manifold $M^n$ has a Lipschitz structure and any two such structure …
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8
votes
Homotopy properties of Lie groups
A very useful fact is that every connected Lie group is rationally homotopy equivalent to the product of several odd dimensional spheres where the number of spherical factors is equal to the rank of t …
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5
votes
on product of some spaces
If $X$ is a smooth manifold (and this is the only case when you can speak of a diffeomorphism between $X\times \mathbb R$ and $\mathbb S^n\times\mathbb R$) then this is true by Poincare. If $X$ is not …
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3
votes
Accepted
Transversal pre-image of a small enough trivial tubular neighborhood contains a trivial tubu...
The condition $\mathcal P(\epsilon)$ is needlessly complicated. It's always true without passing to components.
Let $U$ be a tubular neighborhood of $S$ diffeo to $[-1,1]\times S$. Let $\pi: U\to \mat …
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