Questions tagged [smoothing-theory]

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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 ...
5
votes
1answer
134 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 $R^j\subset R^n$....
2
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0answers
144 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 ...
3
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1answer
181 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 ...
1
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1answer
136 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 $\...
3
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0answers
199 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 ...
22
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2answers
3k 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] \...
12
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1answer
830 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?
18
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1answer
1k views

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 ...