Questions tagged [smooth-structures]
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28 questions
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Explicit construction of the non-trivial element in the mapping class group of 7-sphere
As there are 2 smooth structures on $S^8$, the mapping class group (smooth, orientation-preserving) of $S^7$ is $\mathbb{Z}/2$. I am wondering if there is some explicit construction of the non-trivial ...
- 727
14
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1
answer
636
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Exotic sphere without smooth antipodal map
Let $\iota: S^n \rightarrow S^n$ be the antipodal map. Does there exist an exotic sphere $M^n$ (of any dimension $n$) such that there does not exist a smooth involution $I: M^n \rightarrow M^n$ ...
- 1,558
1
vote
1
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146
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Linear combinations of smooth sections
I am dealing with smoothness issues for which I do not even know a successful approach, so any help or reference would be welcome.
Let $M$ be a manifold, $E\to M$ a smooth vector bundle, and let $\...
- 1,562
1
vote
0
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Smooth structures on codimension-2 Thom-Mather stratifications
Consider a Thom-Mather stratified space with one singular stratum of codimension two:
$$
\rho : M \to [0,\infty), D = \\{ x \in M \mid \rho(x) = 0 \\}.
$$
Since $D$ is codimension two, the real blow-...
2
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107
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Differential equations where we are tryng to find a smooth space(maybe with additional structure)
Is there literature on the differential equation $TM=N$ where $M$ and $N$ are two smooth spaces (maybe with additional structure) and $T$ stands for the tangent functor?
user522448
4
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0
answers
561
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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 ...
- 262
17
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2
answers
2k
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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) - $\mathbb{R}$ ...
- 8,080
4
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187
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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 $U$...
- 20.2k
11
votes
0
answers
288
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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 ...
- 47.3k
2
votes
1
answer
291
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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 ...
- 1,245
12
votes
3
answers
1k
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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 $...
- 545
2
votes
0
answers
379
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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 ...
- 21
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 ...
- 1,946
19
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1
answer
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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 ...
- 28.8k
20
votes
2
answers
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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 ...
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