The `-homology of even Coxeter groups

Given a Coxeter system (W, S), there is an associated CW-complex, denoted Σ(W, S) (or simply Σ), on which W acts properly and cocompactly. This is the Davis complex. L, the nerve of (W, S), is a finite simplicial complex. We prove that when (W, S) is an even Coxeter system and L is a flag triangulation of S, then the reduced `-homology of Σ vanishes in all but the middle dimension. In so doing, our main effort will be examining a certain subspace of Σ called the (S, t)-ruin, for some t ∈ S. To calculate the `-homology of this ruin, we subdivide a component of this ruin into subcomplexes we call colors and then employ a series of Mayer-Vietoris arguments, taking the union of these colors. Once we have established the `-homology of the (S, t)-ruin, we will be able to calculate that of Σ.