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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”.
9
votes
Accepted
How to define relative orientation in terms of (co)homology?
Yes, if I understand the question right.
Reading between the lines, I suppose that you would define an orientation of a smooth $n$-manifold $X$ to be an isomorphism $L\otimes L\to \Omega^n_X$ where $L …
- 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
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
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
10
votes
Accepted
Do $h$-cobordism groups arise from a 'Thom-like' spectrum?
This topic has a very different flavor from what is usually meant by cobordism. No, the correspondence between cobordism classes and homotopy classes (of a Thom space or Thom spectrum) has no analogue …
- 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
4
votes
Accepted
On the proof of the surgery step in Wall's book
The theorem has the hypothesis "$f$ is in this class", meaning that the embedding $f$ is in the regular homotopy class of immersions determined by $F$ together with the element of the relative homotop …
- 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
2
votes
Accepted
Isotopy extension theorem: how non-unique is ambient isotopy
If I interpret the question correctly then the answer is "yes". You seem to be asking whether, if $H'$ is an isotopy satisfying the same conditions as $H$, there must be a one-parameter family of such …
- 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
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
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
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
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
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