Spherical complexes
Sara Faridi, Thiago Holleben
TL;DR
This work introduces spherical complexes—simplicial complexes whose subsequences obtained by links and deletions have the homology of a sphere or are acyclic—and develops a robust combinatorial framework to study their topology and algebraic consequences. By constructing filtrations via induced subcomplexes $\Delta(X\mid Y)$ and defining $d(X\mid Y)$, the authors establish sharp bounds on the possible homology dimensions of these subcomplexes and show the Leray number satisfies $L_K(\Delta)=d_\Delta+1$. They define combinatorial invariants $sign$, $depth$, and $pd$ that parallel classical algebraic invariants, proving a combinatorial Auslander–Buchsbaum identity and connecting these to the Castelnuovo–Mumford regularity via Hochster’s formula. The paper then translates these topological insights into commutative algebra: for a Stanley–Reisner ideal $I_\Delta$, one has $pd(R/I_\Delta)=pd(\Delta)$, $depth(R/I_\Delta)=depth(\Delta)$, and $reg(R/I_\Delta)=depth(\Delta)$ when $\Delta$ is not acyclic, with $L_K(\Delta)=reg(R/I_\Delta)$. These results unify topological and algebraic perspectives, provide tools for computing invariants of independence complexes (e.g., of ternary graphs and simplicial forests), and open avenues related to star packings and Gorenstein/Cohen–Macaulay questions.
Abstract
In this paper we define spherical complexes as simplicial complexes with the property that every subcomplex obtained by a sequence of links and deletions either has trivial homology, or has the homology of a sphere. Examples of such complexes are independence complexes of ternary graphs and independence complexes of simplicial forests. We give criteria for when a spherical complex is acyclic, and describe the dimension of the sphere when it is not. We then apply our results to compute the Leray number of these complexes, and define combinatorial invariants for them which are counterparts to algebraic invariants of their Stanley-Reisner rings.