# Triangulation of manifolds with corners

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.