Questions tagged [smooth-structures]

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When is a level set an immersed submanifold?

Let $M$ be a smooth manifold and $f\in C^\infty(M)$. Let $S:=f^{-1}(\{c\})$ for some $c\in \operatorname{Im}(f)\subseteq\mathbb{R}$. When does exist a manifold $N$ with $\dim(N)<\dim(M)$ and a ...
user1234567890's user avatar
17 votes
2 answers
2k views

Are there only two smooth manifolds with field structure: real numbers and complex numbers?

Is it true that in the category of connected smooth manifolds equipped with a compatible field structure (all six operations are smooth) there are only two objects (up to isomorphism) - $\mathbb{R}$ ...
Arshak Aivazian's user avatar
4 votes
0 answers
155 views

Extension of smooth structure on three dimensional topological manifolds

Let $M$ be a three dimensional compact topological manifold and $U$ an open set in $M$ homeomorphic to some smooth $C^k$ manifold $U'$, $1 \leq k \leq \omega$. Can we extend the smooth structure on $U$...
coudy's user avatar
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11 votes
0 answers
245 views

Almost isometric manifolds are diffeomorphic

I am looking for a reference to the following statement. (It should be known --- I saw it before, don't remember where; search by keywords did not help.) Let $f\colon M\to N$ be a homeomorphism ...
Anton Petrunin's user avatar
2 votes
1 answer
232 views

On the proof of "Mapping space is a Chen space"

According to the page 5 in the paper Convenient Categories of Smooth Spaces https://arxiv.org/pdf/0807.1704.pdf by Baez and Hoffnung, Chen space is defined as follows: (Note:I used different ...
Adittya Chaudhuri's user avatar
12 votes
3 answers
832 views

Can triangulations (or some related combinatorial structure) distinguish smooth structures on $RP^4$?

There are exotic versions of $RP^4$, constructed by Cappell-Shaneson, which are homeomorphic but not diffeomorphic to the standard $RP^4$. One way to distinguish them is via the $\eta$ invariant of $...
Joe's user avatar
  • 535
2 votes
0 answers
226 views

Existence of smooth structures on topological $3$-manifolds with boundary

It is said in this thread Unique smooth structure on $3$-manifolds that every topological $3$-manifold admits a smooth structure. However it is not specified whether the manifolds are allowed to have ...
Dennis's user avatar
  • 21
5 votes
0 answers
120 views

Is there a non-smoothable punctured manifold?

Does there exist a connected topological manifold $M$ such that $M-\{pt\}$ is non-smoothable? My understanding is that Quinn showed that these are always smoothable in dimension 4 (in fact in ...
Cihan's user avatar
  • 1,586
13 votes
1 answer
2k views

Unique smooth structure on 3-manifolds

Do you know a good reference for the existence and uniqueness of a smooth structure on $3$-manifolds? As far as I understand topological $3$-manifolds admit a unique smooth structure. I could find ...
Piotr Hajlasz's user avatar
20 votes
2 answers
1k views

Open subsets of Euclidean space in dimension 5 and higher admitting exotic smooth structures

Up to dimension 3, homeomorphic manifolds are diffeomorphic (in particular, homeomorphic open subsets of $\mathbb{R}^n$ ($n\leq 3$) are diffeomorphic). It is known that there are uncountably many ...
Nautilus's user avatar
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14 votes
1 answer
902 views

Does every dimension $n\geq4$ admit a manifold with an exotic smooth structure?

It is known that $\mathbb{R}^4$ has exotic smooth structures, and there are many such examples in higher dimensions, such as the famous 7-sphere. My (probably very naive) question is, for every $n\...
timur's user avatar
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1 vote
0 answers
74 views

Does a homotopy sphere that bounds a highly connected manifold also bound a parallelizable manifold?

Suppose that the homotopy sphere $\Sigma^{n}$ can be realized as the boundary of a smooth $(n+1)$-dimensional cobordism that is $(n-1)/2$-connected for $n$ odd (respectively, $(n-2)/2$-connected for $...
Rahmpilz's user avatar
  • 165
3 votes
1 answer
323 views

Non-diffeomorphic smooth structures on the quotient of a manifold by an integrable distribution

In geometric quantization, one of the important ingredients is an integrable distribution $D$ (let's say real) on some manifold $M$ (symplectic, but this is not important). The resulting object is $M/...
Alex M.'s user avatar
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17 votes
2 answers
1k views

Exotic smooth structures on Lie groups?

If a topological group $G$ is also a topological manifold, it is well-known (Hilbert's 5th Probelm) that there is a unique analytic structure making it a Lie group. However, for a compact Lie group $...
Piojo's user avatar
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14 votes
3 answers
2k views

When a homeomorphism is a diffeomorphism w.r.t to a suitable smooth structure?

Assume we have a homeomoprhism $\phi:M\rightarrow M$, where $M$ is a topological manifold which admits at least one smooth structure. Is it always possible to construct a smooth structure on $M$ w.r....
Asaf Shachar's user avatar
  • 6,611
2 votes
0 answers
156 views

Relationship between tangent spaces and tangent categories for smooth topoi

Let $\mathscr E$ be a smooth topos, where I am using the terminology of nLab. In particular that imples the existence of a line object $R$ and an "infinitesimally thickened point", which is an object ...
A Rock and a Hard Place's user avatar
3 votes
0 answers
373 views

Is a continuous map between smoothable manifolds of the same dimension always smoothable?

(My question is inspired by this math.SE question, whose negative answer I showed by a dimension-increasing map.) Is it the case that for all smoothable manifolds $M_0$ and $M_1$ with the same ...
user avatar
3 votes
0 answers
501 views

Closed 4-manifolds with uncountably many differentiable structures

I know that $\mathbb{R}^4$ admits uncountably many differentiable structures and I was wandering what happen if we consider closed 4-manifolds. Are there any closed 4-manifolds with uncountably many ...
Dario's user avatar
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33 votes
0 answers
2k views

Do there exist exotic 4-tori?

More precisely: are there known manifolds which are homeomorphic, but not diffeomorphic to the standard 4-torus? Are there any nice invariants distinguishing such manifolds? Related: if such a ...
Jan Jitse Venselaar's user avatar
8 votes
2 answers
605 views

Construction of exotic spheres that do not bound parallelizable manifolds

There are at least two ways to construct homotopy spheres that bound parallelizable manifolds, namely Milnor's plumbing construction and Brieskorn's method of singularities, and each of these methods ...
Ben Knudsen's user avatar
12 votes
1 answer
958 views

Are there non-compact, non-smoothable manifolds?

There do exist manifolds which do not admit any smooth structure at all. But the only examples I've heard of are all compact. Are there any non-compact, non-smoothable manifolds?
AlexE's user avatar
  • 2,926
5 votes
1 answer
239 views

smooth homotopy on exotic R^4

Take an exotic $\mathbb{R}^4$ i.e. $V = (\mathbb{R}^4,d)$ such that $V$ is not diffeomorphic to $\mathbb{R}^4$ with standard metric. Is it true (obvious?) that any two smooth maps $f_1, f_2: S^k \to ...
Maciej Starostka's user avatar
20 votes
5 answers
2k views

Is there a relation between 4-dimensional general relativity and exotic smooth structures on $\mathbb{R}^4$?

Let's say General Relativity is the study of the Einstein equation on smooth Lorentzian manifolds, i.e. pseudo-Riemannian manifolds of signature $(n-1,1)$. I've heard more than once people say that ...
Qfwfq's user avatar
  • 22.7k