Eulerian stratification of polyhedra
A compact polyhedron is a finite union of convex polytopes; a finite (open) polyhedron is the complement of one compact polyhedron in another. A stratification of a compact polyhedron \(X\), indexed by a partially ordered set \(P\), is a decomposition \(X= \bigsqcup_{a\in p}X_a\) of \(X\) into disjoint finite polyhedra \(X_a\), such that for each \(a\in P\) the closure \(\overline{X}_a\) satisfies \[ \overline{X}_a= \bigsqcup_{b\leq a}X_b. \] Further, \(X\) is called Eulerian if, for each \(a\leq b\), \[ \chi(a,b)= \chi(\text{lk} (x,X_b)) \] is independent of the choice of \(x\in X_a\); here, lk is the usual link. Each polyhedron \(X\) is considered here as a triangulation \(\Delta\) consistent with the stratification. Finally, to each \(a\in P\) is attached a weight \(\omega(a)\). The weighted \(f\)-vector is then \[ f(X,\Delta,\omega)= \sum_{a\in P}f(X_a,\Delta_a) \omega(a), \] with \(f= (f_0,f_1,\dots)\) the usual \(f\)-vector. In [Adv. Appl. Math. 19, No. 1, 144-168 (1997; Zbl 0879.52009)] the authors generalized the classical Dehn-Sommerville equations for compact Eulerian manifolds \(M\) with boundary \(\partial M\) in the form \(D(n) f(M,\Delta)= f(\partial M,\partial\Delta)\), with \(D(n)\) the coefficient matrix arising from the usual Dehn-Sommerville equations for simplicial \(d\)-polytopes. They also noted the properties \(D(n-1)D(n)=0\) and \(\chi D(n)= (1-(-1)^n)\chi\). (Here, \(\chi= f_0- f_1+ f_2-\cdots\) is regarded as a linear map on \(f\)-vectors.) In this paper, these results are generalized to stratified Eulerian polyhedra. In particular, \[ D(n) f(X,\Delta,\omega)= f(X,\Delta,\partial_n \omega), \] with \[ \partial_n\omega(a)= [1-(-1)^n] \omega(a)+(-1)^n \sum_{a\leq b}\chi(a,b)\omega(b) \] the \(n\)th boundary weight. Further, the linear conditions on \(f(X,\Delta,\omega)\) for different triangulations \(\Delta\) are described (one is obviously the Euler characteristic). Finally, the numbers \(\chi(a)\) and \(\chi(a,b)\) satisfy \[ \begin{alignedat}{2} \sum_{a\leq b}\chi(a) \chi(a,b) &=0, &&\;\text{ for any fixed }b,\\ \sum_{a\leq c\leq b}\chi(a,c) \chi(c,b) &= 2\chi(a,b), &&\;\text{ for any fixed }a\leq b,\\ \chi(a,a) &=1- (-1)^{d(a)};&& \end{alignedat} \] for this, \(d(a)= \dim X_a\). Modulo some low-dimensional quirkiness (the authors' word, if not their spelling), these conditions are also sufficient for a suitable Eulerian stratified polyhedron \(X\) indexed by \(P\) to exist.
scientific article; zbMATH DE number 1221609
scientific article