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Let's begin with some definitions:

A (smooth) manifold with corners is a Hausdroff (and second countable if you want) space that can be covered by open sets homeomorphic to $\mathbb R^{n-m} \times \mathbb R_{\geq 0}^m$ for some (fixed) $n$ (but $m$ can vary), and such that all transition maps extend to smooth maps on open neighborhoods of each point.

An open $n$-simplex is the interior of a $n$-simplex $\sigma=[v_0,v_1,\ldots, v_n]$, i.e. the set $$\{t_0v_0+ \cdots+ t_nv_n \in \mathbb{R}^{m} \; \colon \; \sum_i{t_i}=1 \; \wedge \; \forall i: t_i> 0 \}$$ where $v_0, v_1 , \ldots, v_n$ are affine independent points in $\mathbb{R}^m$.

A simplicial complex is a finite collection $\mathcal{K}$ of open simplices in $\mathbb{R}^n$, satisfying that given two open simplices in the complex, the intersection of their closures is the empty set or the closure of an open simplex in the complex.

The definition of a simplicial complex above may differ from the standard one. We follow:

Y. Baryshnikov and R. Ghrist. Target enumeration via Euler charac- teristic integrals. SIAM J. Appl. Math. 70.3 (2009), pp. 825–844.

.

L. Van den Dries. Tame topology and o-minimal structures. Vol. 248. London Mathematical Society Lecture Notes Series. Cambridge Univer- sity Press, 1998.

Obviously, usual simplicial complexes are a particular case of simplicial complexes with our definition (that are compact simplicial complexes for us).

My question is:

  • Is there any triangulation result for manifolds with corners involving these simplicial complexes? (or other cellular spaces even?) (topological manifolds with corners of dimension less or equal to three are topological manifolds with boundary (of dimension less or equal to three), so they are triangulable). Therefore I ask for smooth manifolds.

Thanks in advance!

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    $\begingroup$ According to arXiv:math/0611839v2 the answer is "yes" and details can be found in the references of that paper. It seems like the articles in the references, e.g. the paper by Goresky available at www.math.ias.edu/~goresky/pdf/triangulations.pdf work in a general setting of stratified spaces, so one probably has to do some work to understand how to apply such results to manifolds with corners. (I have not looked in any detail at these papers, so I'm not posting this as an answer.) $\endgroup$ – Dan Ramras Sep 24 '17 at 23:35
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    $\begingroup$ The Goresky paper shows that Thom-Mather stratified spaces can be triangulated, so then the question becomes whether manifolds with corners can be Thom-Mather stratified, which seems likely but not completely obvious. Perhaps that's what's in the other references? $\endgroup$ – Greg Friedman Oct 13 '17 at 4:26
  • $\begingroup$ Yes, that is exactly the situation @GregFriedman . In fact, by the time the other comment was made I solved the problem by asking the question to some colleagues but neither I manage to find the time to answer the question nor I found any reference to cite. $\endgroup$ – D1811994 Oct 13 '17 at 19:50

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