On the generalized lower bound conjecture for polytopes and spheres (Q355989)

In 1971, \textit{P. McMullen} and \textit{D. W. Walkup} [Mathematika, Lond. 18, 264--273 (1971; Zbl 0233.52003)] conjectured that the \(h\)-vector of a simplicial \(d\)-polytope satisfies:NEWLINENEWLINE\(h_0 \leq h_1 \leq \cdots \leq h_{\lfloor \frac{d}{2}\rfloor}\) and \(h_{r-1} = h_r\) for some \(1 \leq r \leq \lfloor \frac{d}{2} \rfloor\) if and only if \(P\) is \((r-1)\)-stacked.NEWLINENEWLINEWe say that \(P\) is \((r-1)\)-stacked if \(P\) (i.e., the boundary of \(P\) and its interior) can be triangulated without introducing any new faces whose dimension is smaller than \(d-r\). Thus the classical notion of a stacked polytope corresponds to the family of \(2\)-stacked polytopes.NEWLINENEWLINE\textit{R. P. Stanley} [Adv. Math. 35, 236--238 (1980; Zbl 0427.52006)] showed that the boundary complex of a simplicial \(d\)-polytope has the weak Lefschetz property. Thus not only is the first condition of the McMullen-Walkup conjecture true, but a stronger result holds. Namely, the \(g\)-vector, \(g(P) = (1, h_1-h_0, h_2-h_1, \ldots, h_{\lfloor \frac{d}{2}\rfloor}-h_{\lfloor \frac{d}{2}\rfloor-1})\) is an \(M\)-sequence, meaning it also satisfies Macaulay's nonlinear inequalities that bound the relative growth of its successive entries.NEWLINENEWLINEFor example, the case that \(r=2\) is the classic Lower Bound Theorem of \textit{D. Barnette} [Pac. J. Math. 46, 349--354 (1973; Zbl 0264.52006)] (in the case that \(d=3\) or \(4\)), which was generalized to all dimensions by \textit{G. Kalai} [Invent. Math. 88, 125--151 (1987; Zbl 0624.52004)]. The inequality \(h_1 \leq h_2\), which holds for all simplicial polytopes and more generally for homology manifolds of dimension at least 4, implies that the boundary of a stacked \(d\)-polytope on \(n\) vertices has the componentwise minimal \(h\)-vector among all \((d-1)\)-dimensional homology manifolds on \(n\) vertices. Moreover, \(\Delta\) is a simplicial homology manifold for which \(h_2=h_1\) if and only if \(\Delta\) is the boundary complex of a stacked \(d\)-polytope. The interior of a stacked \(d\)-polytope can be triangulated by only introducing faces of dimension \(d\) or \(d-1\) in a very natural way: a face \(F \subseteq V(\Delta)\) is added to the triangulation if and only if the \((d-2)\)-dimensional skeleton of \(F\) is contained in \(\Delta\).NEWLINENEWLINEThis paper proves the second part of the McMullen-Walkup conjecture. As an extension of the above construction for triangulating stacked polytopes, the authors consider the simplicial complex \(\Delta(i)\) that is obtained from a simplicial complex \(\Delta\) by adding all faces \(F \subseteq V(\Delta)\) whose \(i\)-skeleton is contained in \(\Delta\). They show (Theorem 1.2) that if \(\Delta\) is the boundary of a \(d\)-polytope and \(h_{r-1} = h_r\) for some \(1 \leq r \leq \lfloor \frac{d}{2} \rfloor\), then \(\Delta(d-r)\) is the unique geometric triangulation of \(P\) whose \((d-r)\)-skeleton coincides with that of \(P\). Further, they show (Theorem 1.3) that the same conclusion holds for any simplicial homology sphere that satisfies the weak Lefschetz property. The algebraic \(g\)-conjecture for homology spheres [\textit{E. Swartz}, J. Eur. Math. Soc. (JEMS) 11, No. 3, 449--485 (2009; Zbl 1167.52013)] posits that any simplicial homology sphere satisfies the weak Lefschetz condition; so if the \(g\)-theorem is true, then Theorem 1.3 will hold for the much broader class of simplicial homology spheres.