Face numbers of pseudomanifolds with isolated singularities (Q2890396)

Let \({\Bbbk}\) be a field. A \textit{\({\Bbbk}\)-pseudomanifold with isolated singularities} is a (finite) pure simplicial complex \(\Delta\) in which the links of vertices are \(\Bbbk\)-homology \((\ell-1)\)-manifolds, or equivalently, the links of edges are \({\Bbbk}\)-homology \((\ell-2)\)-spheres, \(\ell=\dim(\Delta)\). This is a partial generalization of \textit{\(\Bbbk\)-Buchsbaum complex}, in which the vertex links are Cohen-Macaulay over \(\Bbbk\), and is a special case of \textit{\(\Bbbk\)-space with isolated singularities}, in which the vertex links are \(\Bbbk\)-Buchsbaum. The \(f\)-vector \(f=(f_0, \dots, f_\ell)\) of \(\Delta\) records the number \(f_i\) of simplices of each dimension \(i\); the \(h\)-vector \((h_0, \dots, h_d)\), \(d=\ell+1\), is obtained from \(f\) by a linear change of coordinates. The face ring \(\Bbbk(\Delta)\) of \(\Delta\) is the quotient of the polynomial ring on the vertices by the ideal generated by monomials corresponding to non-simplices of \(\Delta\). In the paper under review, the authors extend fundamental results about \(h\)-vectors and face rings of rational convex polytopes, triangulated spheres, and Cohen-Macaulay and Buchsbaum complexes, to pseudomanifolds and \(\Bbbk\)-spaces with isolated singularities.NEWLINENEWLINEThe first result is a generalization of the Dehn-Sommerville relations to pseudomanifolds with isolated singularities. The original relations \(h_{d-i}=h_i\) for spheres extend to homology manifolds with a correction term involving \(\chi(\Delta)\); the authors' generalization includes a further correction term involving Euler characteristics of links of singular vertices, which they then observe can also be expressed in terms of \(\chi(\Delta)\).NEWLINENEWLINEThe other results in the paper require that singularities be \textit{homologically isolated}, meaning that the local cohomology spaces \(H^i(|\Delta|,|\Delta|-p; \, \Bbbk)\) at the singular points \(p\) map to linearly independent subspaces in \(\tilde{H}^i(|\Delta|,\Bbbk)\), for \(i<\ell\). The authors compute the Hilbert series \(\sum_{i=0}^d h_i't^i\) of an arbitrary Artinian reduction of \(\Bbbk[\Delta]\), for \(\Delta\) a \(\Bbbk\)-space with homologically isolated singularities, generalizing a formula of Schenzel for Buchsbaum complexes. The difference \(h_i'-h_i\) is shown to be a PL-topological invariant of \(|\Delta|\). The result is used to prove lower and upper bound theorems. The authors establish a lower bound on \(g_2=h_2-h_1\) in case \(d\geq 5\) or \(d=4\) and there are at most five singular points, generalizing their own earlier result for homology manifolds. The Upper Bound Conjecture \(h_i\leq \binom{n-d+i-1}{i}\) is established for Eulerian pseudomanifolds with homologically isolated singularities and at least \(3d-4\) vertices. (A complex is \textit{Eulerian} if the Euler characteristic of the link of any simplex, and of \(\Delta\) itself, is equal to that of a sphere of the same dimension.) This upper bound was previously known for all Eulerian homology manifolds, and for arbitrary Eulerian complexes with at least \(O(d^2)\) vertices. In the last section the authors apply their results to characterize the \(f\)-vectors of triangulations of three interesting examples from the literature.