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Results tagged with differential-topology
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user 6666
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”.
73
votes
Accepted
Example of a manifold which is not a homogeneous space of any Lie group
$\pi_2$ of a Lie group is trivial, so $\pi_2(G/H)$ is isomorphic to a subgroup of $\pi_1(H)$, which is finitely generated (isomorphic to $\pi_1$ of a maximal compact subgroup of the identity component …
- 55.9k
70
votes
How can there be topological 4-manifolds with no differentiable structure?
The usual convolution method for approximating continuous maps by smooth maps does not succeed in approximating invertible [resp. injective] continuous maps by invertible [resp. injective] smooth map …
- 55.9k
61
votes
Accepted
Do rings of smooth functions differ from rings of continuous functions?
No. In both the smooth function ring and the continuous function ring a maximal ideal $\frak m$ consists of the functions vanishing at some point. In the smooth case $\frak m/\frak m^2$ is the cotange …
- 55.9k
47
votes
Accepted
When is there a submersion from a sphere into a sphere?
In most cases $\pi_{n+k}(S^k)$ is a finite group, so that the homotopy fiber of any map $S^{n+k}\to S^k$ is rationally equivalent to $\Omega S^k\times S^{n+k}$ and therefore has homology in arbitraril …
- 55.9k
33
votes
Accepted
"Largest" finite-dimensional Lie subgroups of Diff(S^n), are they known?
You can make big Lie groups act effectively on small manifolds by cheating: make the group a product of groups, with each factor acting by compactly supported diffeomorphisms on a different disjoint o …
- 55.9k
26
votes
Accepted
Extending a diffeomorphism of the sphere $S^2$ to the ball $D^3$
More a survey of related things than an answer, but here goes.
Let's write $D(n)$ for the space of compactly supported diffeomorphisms $\mathbb R^n\to \mathbb R^n$. A reasonable guess might be that t …
- 55.9k
25
votes
Is there a sheaf theoretical characterization of a differentiable manifold?
For another perspective, think about definitions of "complex manifold" or "real analytic manifold". Normally you use atlases for this, imposing the Hausdorff and paracompactness conditions separately. …
- 55.9k
21
votes
Manifolds with polynomial transition maps
Brief sketch of slight simplification of Bryant's answer:
Without loss of generality $\mathcal A$ is maximal with respect to the condition that all transition functions are polynomial. Now make a spa …
- 55.9k
21
votes
Piecewise-smooth manifolds?
A homeomorphism $h:U\rightarrow V$ between open subsets of $\mathbb R^n$ is called piecewise differentiable (PD) -- you could also say piecewise smooth -- if there is a triangulation of $U$ by linear …
- 55.9k
21
votes
Accepted
Sum of two tangent bundles of $S^{2n}$
Yes. Let $V$ be a real vector bundle whose base is a $d$-dimensional manifold or cell complex, and whose fibers are $r$-dimensional. Then (1) if $r>d$ then $V=W\oplus \epsilon$ where $\epsilon$ is a t …
- 55.9k
17
votes
Accepted
Homotopy of space of immersions, Smale-Hirsch theorem
No. For example, if $M$ is a Moebius band then, at least for even $k$, $Imm(M,\mathbb R^{2+k})$ is not homotopy equivalent to $Imm(S^1\times \mathbb R,\mathbb R^{2+k})$.
The latter is equivalent to …
- 55.9k
15
votes
Accepted
Does an oriented $S^3$ fiber bundle admit the structure of a principal $SU(2)$-bundle?
No. (The main idea here is present in Dylan Wilson's comment.)
Every principal $SU(2)$-bundle over $S^2$ is trivial, because $\pi_1 SU(2)$ is trivial. But there is a nontrivial oriented bundle over $ …
- 55.9k
14
votes
Accepted
Possible mistake in Cohen notes "Immersions of manifolds and homotopy theory" (version 27 Ma...
The statement is false in two ways. First, two immersions might not even be homotopic even though their normal bundles are both, say, trivial. Second, even if $M=\mathbb R^m$, regular homotopy classes …
- 55.9k
13
votes
Accepted
What is the status of the PL-pseudoisotopy stability theorem?
As you say, Hatcher once argued that the map $\sigma_M^{PL}:C^{PL}(M)\to C^{PL}(M\times I)$ is $k$-connected where $k$ is roughly $n/3$, but the proof was not all there.
And as you say Igusa later pr …
- 55.9k
13
votes
Accepted
Is a manifold-with-boundary with given interior and non-empty boundary essentially unique?
No, there are examples detected by Whitehead torsion. If $P$ is a compact connected $(n-1)$-manifold with empty boundary, then (assuming $n\ge 6$) for every element $\tau$ of the Whitehead group of $\ …
- 55.9k