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Questions tagged [differential-topology]

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”.

1745 questions
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3 answers

Can analysis detect torsion in cohomology?

Take, for example, the Klein bottle K. Its de Rham cohomology with coefficients in $\mathbb{R}$ is $\mathbb{R}$ in dimension 1, while its singular cohomology with coefficients in $\mathbb{Z}$ is $\mathbb{Z} \times \mathbb{Z}_2$ in dimension 1. It…
Paul Siegel
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21
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3 answers

Is every linear functional on a smooth finite dimensional vector space automatically smooth?

By a smooth finite dimensional vector space I mean a smooth manifold $M$ together with smooth operations $+ : M \times M \rightarrow M$ and $\cdot : \mathbb{R} \times M \rightarrow M$ turning $M$ into a finite dimensional vector space. Is any…
Tristan Bice
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18
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Perturbation of a smooth manifold and transversality

Let $M$ be a compact smooth manifold and $N$ be a compact smooth submanifold of $M$. The usual transversality theorem claims that for a generic diffeomorphism $f$ of $M$, the submanifolds $N$ and $f(N)$ are transverse. I am interested in another…
user119986
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14
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2 answers

Relation between combinatorial manifolds and PL manifolds

In W. W. Boone, W. Haken, and V. Poenaru, On Recursively Unsolvable Problems in Topology and Their Classification, Contributions to Mathematical Logic (H. Arnold Schmidt, K. Schütte, and H. J. Thiele, eds.), North-Holland, Amsterdam, 1968. a…
Malte
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13
votes
2 answers

Smooth representatives for elements of $\pi_7(\text{exotic $S^7$})$

Let $M$ be $S^7$ with an exotic smooth structure. Since one can smoothen maps, there exist smooth maps $f:S^7\to M$ which are homotopic to the identity (relative to a base point, if you want). Can one make explicit one such map? Can such a map be…
Mariano Suárez-Álvarez
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12
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1 answer

Piecewise-smooth manifolds?

We know about the existence of topological (Top), differentiable (Diff) and piecewise-linear (PL) manifolds, and such things that, say, in four dimensions PL=Diff, but $\ne$Top. The question is: do there exist piecewise-smooth manifolds? Are they…
John Vrem
  • 374
11
votes
2 answers

When is a manifold a tangent bundle?

Given a (smooth) manifold $M$, are there any sufficient, intrinsic properties that would tell you there exists a (smooth) manifold $N$ such that $M$ is diffeomorphic to $TN?$ There are some obvious necessary conditions (like $M$ needs to be…
Jon Cohen
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10
votes
1 answer

Ehresmann's fibration theorem in the C1 class

I have seen on the French Wikipedia that Ehresmann's fibration theorem is stated with the assumption that everything is $C^2$, see Théorème de Ehresmann. (On the English Wikipedia, the assumption is smooth, which I suppose means $C^\infty$, see…
Arnaud Chéritat
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10
votes
1 answer

Examples of sphere bundles

For certain values of $k$ it is known that $\mbox{Diff}(S^k)$ is not homotopy equivalent to $O(k+1)$. So there are sphere bundles that do not arise from vector bundles. Since I've never (knowingly) come across such a sphere bundle I'm interested in…
Dave
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9
votes
2 answers

Finite type vs. finite dimensional cohomology?

I am not quite sure about the terminology, but let's call a $n$-dimensional manifold of finite type if it has a finite open cover $U_1,\ldots,U_k$ such that all intersections are either empty or diffeomorphic to ${\mathbb R}^n$. Every compact…
Andrei Moroianu
  • 2,151
9
votes
1 answer

Can any path in the diffeomorphism group of a smooth compact manifold be approximated by a smooth path?

Given a smooth compact $n$-dimensional manifold $M^{n}$, let $\operatorname{Diff}(M)$ denote the group of smooth diffeomorphisms $M \rightarrow M$ equipped with the Whitney $C^{\infty}$-topology. Let $h_{\phi} \colon [0, 1] \times M \rightarrow M$…
Rahmpilz
  • 165
8
votes
0 answers

Is this invariant of an exotic 4-sphere trivial?

This is a rather naive attempt to construct an invariant of an exotic 4-sphere. Apparently, the lack of useful invariants in this context is a well known issue. This particular invariant is somewhat obvious which probably means it is useless…
Alex Gavrilov
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8
votes
2 answers

Triangulations of exotic 4-spheres

Are there explicit examples of triangulations of exotic 4-spheres?
John Vrem
  • 374
8
votes
1 answer

Why are order-k differential forms sections of the kth exterior power of the cotangent bundle?

The question I ask is in the title. This should be quite well-known, and in fact probably I am going to get the response that it is the definition. To convey my confusion, I have to convey my understanding of what is a differential form and what is…
Akela
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8
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Embedding of parallelizable closed smooth manifold

It seems that any closed parallelizable smooth $n$-manifold can be immersed in the $(n+1)$-dimensional Euclidean space (compared to the general $(2n-1)$-dimensional case). Is there any analogous result for embedding?
Kds
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