A Method for Integral Cohomology of Posets

Homotopy type of partially ordered sets (poset for short) play a crucial role in algebraic topology. In fact, every space is weakly equivalent to a simplicial complex which, of course, can be considered as a poset. Posets also arise in more specific contexts as homological decompositions [10, 6, 16, 20] and subgroups complexes associated to finite groups [7, 26, 4]. The easy structure of a poset has led to the development of several tools to study their homotopy type, including the remarkable Quillen’s theorems [25] and their equivariant versions by Thévenaz and Webb [28]. In spite of their apparent simplicity posets are the heart of many celebrated problems: Webb’s conjecture (proven in [27] and generalized in [21]), the unresolved Quillen’s conjecture on the p-subgroup complex (see [1]), or the fundamental Alperin’s conjecture (see [19]). In this paper we propose a method to compute integral cohomology of posets. This toolbox will be applicable as soon as the poset has certain local properties. More precisely, we will require certain structure on the category under each object of the poset. By means of homological algebra of functors we prove that, in the presence of these local structures, the cohomology of the poset is that of a co-chain complex