Questions tagged [smooth-structures]
The smooth-structures tag has no usage guidance.
23
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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 ...
17
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2
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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) - $\mathbb{R}$ ...
4
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155
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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 $U$...
11
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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 ...
2
votes
1
answer
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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 ...
12
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3
answers
832
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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 $...
2
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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 ...
5
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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 ...
13
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1
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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 ...
20
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2
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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 ...
14
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1
answer
902
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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\...
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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 $...
3
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1
answer
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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/...
17
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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 $...
14
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3
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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....
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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 ...
3
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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 ...
3
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0
answers
501
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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 ...
33
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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 ...
8
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2
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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 ...
12
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1
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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?
5
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1
answer
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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 ...
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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 ...