Cohomology of non-finite CL-shellable posets

Abstract

Shellable complexes are homotopy equivalent to a wedge of spheres of possibly different dimensions, so that the (co)homology of the constant functor over the complex is concentrated in those degrees. In this work, we introduce the concept of a stable functor -a local weakening of fibrancy- over a shellable poset, which ensures the vanishing of the (co)homology of such a functor in specific degrees. The methods are based on a model category structure on the category of functors indexed by a filtered poset and the combinatorial structure of shellable posets. Our techniques work over non-finite and non-pure posets and employ a description of (co)homology via explicit fibrant replacements. Applications include acyclicity criteria for Mackey functors, computation of cohomology of $j$-th exterior powers over arrangement lattices, and homological decompositions for Bianchi groups $Γ_d$ for $d=1,2$.