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

$\DeclareMathOperator\STop{STop}$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 $R^j\subset R^n$....
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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 ...
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### 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 ...
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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 $\... 0answers 199 views ### smoothing a current Let$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$is a smooth ... 2answers 3k views ### Isotopy extension theorems I'm looking for the origins of the isotopy extension theorem in categories other than the smooth category. Precisely, in the smooth category, the isotopy extension theorem says that if$f : [0,1] \...
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 ...