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Given a Coxeter system (W, S), there is an associated contractible CW-complex, Sigma(W, S) (or simply Sigma), on which W acts properly and cocompactly. This is the Davis complex. Denote by L the nerve of (W, S), it is a finite simplicial complex.MoreGiven a Coxeter system (W, S), there is an associated contractible CW-complex, Sigma(W, S) (or simply Sigma), on which W acts properly and cocompactly. This is the Davis complex. Denote by L the nerve of (W, S), it is a finite simplicial complex. Sigma admits a cellulation under which the nerve of each vertex is L.

It follows that if L is a triangulation of Sn-1 , then Sigma is a contractible n-manifold. We prove two results, each a special case of Singers conjecture on the vanishing of the reduced &ell-2-homology of manifolds in all but the middle dimension: (1) If L is a triangulation of S2 , then the reduced &ell-2-homology of Sigma, H* (Sigma), vanishes in every dimension- and (2) If (W, S) is an even Coxeter system and L is a flag triangulation of S3 , then Hi (Sigma) = 0 for i ≠ 2. We prove result (1) using a theorem of Andreev, which gives the necessary and sufficient conditions for a classical reflection group to act on H3 .

Next, we show that result (2) follows from (1) and a series of Mayer-Vietoris arguments. In proving (2), our main effort will be examining a certain subspace of Sigma called the (S, t)-ruin, for some vertex t of L. We subdivide a component of this ruin into subcomplexes we call boundary collars and then use Mayer-Vietoris sequences to calculate the &ell-2-homology of the (S, t)-ruin.

Having done this, we are able to calculate H* (Sigma).

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