Strict \(C^1\)-triangulations in o-minimal structures (Q1741799)

The authors prove a \(C^1\)-triangulation theorem in o-minimal structures. Let \(R\) be a real closed field and consider any o-minimal expansion of \(R\). It is shown that for any closed bounded definable subset \(A\) of \(R^n\) and any finite collection of definable subsets \(B_1,\ldots,B_r\) of \(A\) there exists a definable triangulation \(h : | \mathcal K | \to A\) of \(A\) compatible with \(B_1,\ldots,B_r\), where \(\mathcal K\) is a simplicial complex in \(R^n\), \(h\) is a \(C^1\)-embedding of each simplex \(\Delta \in \mathcal K\), and \(h\) extends to a definable \(C^1\)-mapping defined on a definable open neighborhood of the polyhedron \(|\mathcal K|\) in \(R^n\). This theorem improves a recent result of \textit{T. Ohmoto} and \textit{M. Shiota} [J. Topol. 10, No. 3, 765--775 (2017; Zbl 1376.14060)] in several different aspects; also the method of proof is different. The proof proceeds in two steps. First the existence of a definable \(C^1\)-triangulation of \(A\) compatible with \(B_1,\ldots,B_r\), where \(h : | \mathcal K | \to R^n\) is Lipschitz and \(\{h|_\Delta : \Delta \in \mathcal K\}\) is a \(C^1\)-stratification of \(h\) satisfying Whitney's condition (A), is shown. Then \(h\) is modified in some tubular neighborhoods of simplices in order to obtain a \(C^1\)-extension to a definable open neighborhood of \(|\mathcal K|\).