On Modular Homology of Simplicial Complexes: Saturation

Abstract

Among shellable complexes a certain class is shown to have maximal modular homology, and these are the so-called saturated complexes. We show that certain conditions on the links of the complex imply saturation. We prove that Coxeter complexes and buildings are saturated. © 2002 Elsevier Science (USA)

Saturated simplicial complexes

“…It follows closely our papers [18,19] and it may be useful to consult these papers for further details. However, we hope that the notes in the following section will render this paper reasonably self-contained.…”

“…In [19] we have introduced the notion of saturation for a shellable complex in relation to the prime p. There we already showed that saturation for shellable complexes has several equivalent definitions. The purpose of this section is to expand further on the combinatorial and algebraic significance of saturation.…”

“…We recall the main definitions from [18,19] on modular homology. Assume that ⊆ 2 is a complex of dimension n − 1.…”

On Modular Homology of Simplicial Complexes: Shellability

“…Using the results of this paper it is possible to determine the modular homology of finite buildings. This is the subject of a forthcoming paper [12].…”

On Modular Homology in the Boolean Algebra, III

“…In [1] this situation has been analyzed completely when r = 1: various irreducible S n -representations can be realized in this fashion, and indeed whole inductive systems for symmetric groups arise in this way, for arbitrary p. In two recent papers [12,13] it is shown that these modules play a fundamental role for the modular homology of simplicial complexes in general and for shellable complexes in particular. Our interest here is partly guided by the fact, that in the geometrical setting rank selected posets are important, and this leads to the consideration of r-step maps for r > 1.…”