For the purposes of this question, let's say that a blurring of a smooth triangulation $T$ of a smooth manifold $X$ is a smooth homotopy $h\colon [0,1] \times X \to X$ such that $h_0=\operatorname{id}_X$, $h_t$ maps each simplex of $T$ to itself, and $h_1$ maps some open neighborhood of each simplex $σ$ of $T$ to $σ$.
As an example, consider the $n$-sphere $S^n=\left\{x∈\mathbb{R}^{n+1}:‖x‖=1\right\}$ triangulated into $2^{n+1}$ simplices by the coordinate hyperplanes $x_i=0$. A blurring for this triangulation can be constructed as $h_t(x)=\operatorname{proj}\left(C_t(x)\right)$, where $\operatorname{proj}(x)=x/‖x‖$, $C_t(x_0,…,x_n)=(c_t(x_0),…,c_t(x_n))$, $c_t(r)=(1−t)r+t·b(r)$, and $b$ is a smooth bump function such that $b=\operatorname{id}$ on $(−∞,−ϵ]∪[ϵ,∞)$ and $b=0$ on $[−ϵ/2,ϵ/2]$, where $ϵ<1/(n+1)$.
Does every smooth triangulation of a smooth manifold admit a blurring?
Madsen and Weiss in §A.1 of their paper on Mumford's conjecture incorporate the above notion of blurring into their notion of an extended triangulation (which also involves a total ordering on vertices of $T$, an extension of simplices of $T$ to extended simplices in a compatible way, and a requirement that $h_t$ preserves extended simplices). They assert without proof that any smooth triangulation can be extended to an extended triangulation, which in particular would imply a positive answer to the above question. This assertion is an important step in their proof of Mumford's conjecture.
Some obvious approaches to constructing blurrings fail for various reasons.
For example, one can construct a family of smooth maps $b_n\colon Δ^n→Δ^n$, functorial in the simplex $n$, such that $b_n$ maps an open neighborhood of $∂Δ^n$ to $∂Δ^n$. Using a linear interpolation between $b_n$ and the identity map on $Δ^n$, one can construct a continuous homotopy $h$ that satisfies all the properties of a blurring except smoothness. However, such an $h$ is not smooth on simplices of codimension 1 and higher.
Another approach tries to construct $h$ by induction on the skeleta of $T$. For example, given a map $h \colon [0,1]×T_n→T_n$ defined on the $n$-skeleton $T_n$ of $T$ satisfying the above properties, one could try to extend it to the $(n+1)$-skeleton $T_{n+1}$ of $T$ using (very roughly) the following three steps:
(a) for each $n$-simplex $σ$ of $T$ construct an extension of $h$ to some open neighborhood of the interior of $σ$ inside $T_{n+1}$;
(b) assemble all maps constructed in part (a) into a single extension of $h$ to some open neighborhood of $T_n$ inside $T_{n+1}$;
(c) extend the map in (b) from the open neighborhood of $T_n$ to $T_{n+1}$. Even if one succeeds at (a) and (c), it is not at all clear to me what to do about (b), because there is no reason why all the individual extensions should be compatible near the $(n−1)$-simplices of $T$.
I'm also open to imposing additional conditions on smooth triangulations, as long as one can show that any smooth manifold admits a smooth triangulation with such additional properties.
