  Associated to any finite flag complex L there is a right-angled Coxeter group
W_L and a cubical complex \Sigma_L on which W_L acts properly and cocompactly.
Its two most salient features are that (1) the link of each vertex of \Sigma_L
is L and (2) \Sigma_L is contractible. It follows that if L is a triangulation
of S^{n-1}, then \Sigma_L is a contractible n-manifold. We describe a program
for proving the Singer Conjecture (on the vanishing of the reduced L^2-homology
except in the middle dimension) in the case of \Sigma_L where L is a
triangulation of S^{n-1}. The program succeeds when n < 5. This implies the
Charney-Davis Conjecture on flag triangulations of S^3. It also implies the
following special case of the Hopf-Chern Conjecture: every closed 4-manifold
with a nonpositively curved, piecewise Euclidean, cubical structure has
nonnegative Euler characteristic. Our methods suggest the following
generalization of the Singer Conjecture.
  Conjecture: If a discrete group G acts properly on a contractible n-manifold,
then its L^2-Betti numbers b_i^{(2)} (G)$ vanish for i>n/2.
