Questions tagged [smoothing-theory]

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What heuristic arguments support Montgomery's conjecture for primes in short intervals?

I have a question regarding a conjecture due to H. L. Montgomery on the number of primes in short intervals. The conjecture apparently arises from probabilistic reasoning upon assuming the Riemann ...
AfterMath's user avatar
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5 votes
0 answers
129 views

Smoothing tame topological knots, from an analytic perspective

A tame topological knot (for the purpose of this question) is a topological embedding of $S^1 \times D^2$ into $\Bbb R^3$. Tame topological knots are known to be isotopic to smooth knots. This ...
Ryan Budney's user avatar
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2 votes
1 answer
238 views

Markov triples and Newton-Okounkov bodies of $\mathbb{P}^2$

I am working on symplectic geometry and I have some questions about a degeneration of $\mathbb{P}^2$. Question: Can we obtain the moment polytope (or the polytope associated with the anti-canonical ...
Yunhyung Cho's user avatar
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6 votes
1 answer
511 views

Knots: locally flat, PL and smooth

In the classical dimension (knots in $S^3$), it is considered standard (I think?) that the following sets are in bijective correspondence: locally flat knots up to ambient isotopy; PL-knots up to PL ...
Anthony Conway's user avatar
11 votes
1 answer
409 views

Haefliger trefoil $S^3\hookrightarrow S^6$

It is known that the Haefliger trefoil $S^3\hookrightarrow S^6$ is PL trivial but non-trivial smoothly. I wonder, where exactly does the problem come? Consider its tubular neighborhood $T\cong S^3\...
Victor's user avatar
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1 vote
1 answer
173 views

Is $\operatorname{PL}_{n,n-1}$ contractible?

$\DeclareMathOperator\PL{PL}$Consider the group $\PL_{n,n-1}$ of orientation preserving PL self-homeomorphisms of $\mathbb R^n$ that also preserve $\mathbb R^{n-1}$ pointwise. It is usually understood ...
Victor's user avatar
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5 votes
1 answer
167 views

$\operatorname{STop}_{n,n-2}\simeq S^1$?

$\DeclareMathOperator\STop{STop}$I am interested in any information about the homotopy type of the groups $\STop_{n,j}$ of homeomorphisms of $R^n$ preserving orientation and pointwise $R^j\subset R^n$....
Victor's user avatar
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2 votes
0 answers
225 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
3 votes
1 answer
256 views

Smoothing a periodic function of two variables

Let $F \colon \mathbb{R}^2 \to \mathbb{R}^n$ be a $C^{1}$-function 1-periodic in each variable, so it can be considered as a function on the flat torus $\mathbb{T}^2 = \mathbb{R}^2 / \mathbb{Z}^2$. We ...
demolishka's user avatar
1 vote
1 answer
170 views

log-convexity of Mollified function?

Let $f:{\mathbb R}\rightarrow{\mathbb R}_+$ be a log-convex function. Suppose that $f_{\epsilon}$ is the smoothed version of $f$: $$f_{\epsilon}(x)=\int \varphi_{\epsilon}(x-y)f(y)dy,$$ where $\...
Nigel's user avatar
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3 votes
0 answers
222 views

smoothing a current

Let $M$ be a smooth oriented manifold of dimension $n$ and $T$ a current of dimension $k$ on $M$. Let $\phi:P\times M \to M$ be a proper smooth family of diffeomorphisms of $M$ (i.e. $P$ is a smooth ...
Joe Fu's user avatar
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23 votes
2 answers
4k views

Isotopy extension theorems

I'm looking for the origins of the isotopy extension theorem in categories other than the smooth category. Precisely, in the smooth category, the isotopy extension theorem says that if $f : [0,1] \...
Ryan Budney's user avatar
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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
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18 votes
1 answer
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Diffeomorphisms vs homeomorphisms of 3-manifolds

For a smooth 3-manifold $M$, is the natural map from the space of diffeomorphisms of $M$ to the space of homeomorphisms of $M$, $${\sf Diff}(M) \longrightarrow {\sf Top}(M),$$ a weak homotopy ...
John Francis's user avatar