Elementary moves on triangulations (Q2498939)

In this paper, an \(n\)-polyhedron means a finite union of \(n\)-dimensional polytopes in \(\mathbb{R}^n\). A triangulation of an \(n\)-polyhedron \(P\) is a finite set of \(n\)-simplices, such that the intersection of any two is of dimension at most \(n-1\), and the union of the simplices is \(P\). Note that the intersection of two simplices need not be a face of either. An elementary move is either the dissection of an \(n\)-simplex \(T\) into two \(n\)-simplices by a hyperplane containing an \((n-2)\)-face of \(T\) or the merging of two \(n\)-simplices whose union is an \(n\)-simplex. The main result of the paper is that any triangulation of a polyhedron \(P\) can be obtained from any other triangulation of \(P\) by a finite sequence of elementary moves. As a corollary the authors show that every valuation defined on the set of simplices has a unique extension to a valuation on the set of polyhedra. Related classical results assume face-to-face subdivisions and allow for simplicial isomorphism. See \textit{W. B. R. Lickorish} [Geom. Topol. Monogr. 2, 299--320 (1999; Zbl 0963.57013)].