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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”.
12
votes
Accepted
Finite type vs. finite dimensional cohomology?
It does matter what kind of cohomology you use:
First think of the connected sum of an infinite number of real projective spaces end to end. De Rham cohomology cannot tell that these were not sphere …
- 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
3
votes
On a parallelizable manifold, is there always a frame satisfying $[X_i,X_j]=0$?
If $M$ is connected and non-compact of dimension $m$, then parallelizability implies that $M$ can be immersed in $\mathbb R^m$, and this implies existence of such a framing.
- 55.9k
6
votes
Accepted
Quotient of trivial bundles
If the dimension of the base $X$ of the bundles is less than the difference $n-k$ of the fiber dimensions then the quotient bundle is trivial. To see this it is enough to consider the case $k=1$ and t …
- 55.9k
12
votes
Are there good product rules on the $k$-sphere?
$S^0$, $S^1$, and $S^3$ have well-known smooth group structures. These can be obtained from well-known bilinear multiplications in $\mathbb R$, $\mathbb R^2$, and $\mathbb R^4$. $S^7$ does not have a …
- 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
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
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
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
3
votes
Accepted
Decomposing proper map into closed embedding and proper submersion
Yes. Choose a smooth (but not necessarily closed) embedding $i:X\to W$ where the manifold $W$ is compact, for example a sphere. Together $i$ and $f$ give a smooth map $X\to W\times Y$ that is both pro …
- 55.9k
6
votes
Accepted
Connectivity of the space of transverse vector fields
Let $F(M)$ be this space of vector fields. Let's work this out for an even-dimensional sphere $M=S^n=S^{2p}$, so $n=2p$ and $k=2p-1$. I claim that it is not rationally $3$-connected, if $n\ge 3$.
This …
- 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
1
vote
Generalising the parametric transversality theorem to a foliation
The theorem says that if $F:M\times S\to N$ is transverse to $R$ then for almost every $s\in S$ the map $\phi_s:M\to N$ given by $m\mapsto F(m,s)$ is transverse to $R$. ($S$ being connected is irrelev …
- 55.9k
4
votes
Which submanifolds are zero sets of $\mathbb{R}^n$-valued maps?
If $N$ is framed and of codimension $n>0$ then a normal vector field gives you a manifold $N'$ isomorphic to $N$ (near $N$, running parallel to $N$). For $N$ to be of the form $f^{-1}(0)$ with $f:M\to …
- 55.9k
7
votes
Existence of sections of the evaluation map for the diffeomorphism group
To add to Andy Putman's answer: An interesting modification of the question is to replace diffeomorphisms by homotopy equivalences. Thus, for a given based manifold or CW complex $(X,p)$, is there a c …
- 55.9k