Questions tagged [smoothing-theory]

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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 ...
3
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1answer
136 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
122 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
195 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 ...
20
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2answers
2k 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
votes
1answer
757 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 ...