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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.

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    $\begingroup$ I am trying to picture this. Cut up a tennis ball into octants, through the coordinate planes through the centre of the ball. Use those octants as my triangulation. This $h_t$ takes each octant to itself, I think, for each $t$. But then so does $h_1$. So $h_1$ doesn't map any point outside a given octant into that octant, because that point belongs to the other octants. But then $h_1$ doesn't map a neighborhood of our given octant to that octant. Where am I lost? $\endgroup$ – Ben McKay Mar 30 '16 at 18:46
  • $\begingroup$ @BenMcKay: The problem lies with the last sentence (“But then“). What's happening here is that the map h_1 collapses a neighborhood of each coordinate plane to that plane. In your example, h_t can be constructed explicitly: choose a cutoff function c that vanishes in a small neighborhood of 0 and is identity outside of a larger neighborhood. Apply c to all three coordinates of a point in the sphere and then project back to the sphere using a radial projection. This gives you h_1, and h_t can be constructed by linearly interpolating between c and id. $\endgroup$ – Dmitri Pavlov Mar 31 '16 at 7:44
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    $\begingroup$ @JesseC.McKeown: What you're suggesting is the first approach discussed in the main post (see the paragraph that starts with “For example, one can construct a family of smooth maps…”). It doesn't work because no matter what the maps b_n are (S_n-equivariant or not), the map h_1 will not be smooth on simplices of codimension 1 and higher (if the ambient manifold has dimension 2 or higher). $\endgroup$ – Dmitri Pavlov Mar 31 '16 at 17:05
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    $\begingroup$ @JesseC.McKeown: The entire family of approaches cannot work: no matter which blurrings Δ^n→Δ^n you choose (functorial in n or not, S_n-equivariant or not), the resulting blurring of the entire triangulation will be smooth away from simplices of codimension 1, but only continuous (and not smooth) on simplices of codimension 1. Even more strongly: for any fixed n≥2 and for any smooth map f: Δ^n→Δ^n one can construct a smooth triangulation T with two adjacent n-simplices a and b such that the map induced by f on T is not smooth on the intersection of a and b. $\endgroup$ – Dmitri Pavlov Mar 31 '16 at 18:00
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    $\begingroup$ @JesseC.McKeown: With respect to your last proposal, it is unclear how to construct “normal scalings around each facet”. In private correspondence with somebody else it was suggested to me that the existence of such scalings can be included in the definition of a smooth triangulation (the term “conical triangulation” was used to describe such objects). However, this simply moves the problem to a different place: we still have to prove that any smooth manifold admits a conical smooth triangulation. $\endgroup$ – Dmitri Pavlov Apr 2 '16 at 14:31

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