Even- vs. Odd-dimensional Charney-Davis Conjecture

More than once we have heard that the Charney-Davis Conjecture makes sense only for odd-dimensional spheres. This is to point out that in fact it is also a statement about even-dimensional spheres. A conjecture of Heinz Hopf asserts that the sign of the Euler characteristic of a smooth Riemannian 2d-dimensional manifold of non-positive sectional curvature is the same for all such manifolds, that is, the same as that of product of non-positively curved surfaces: (−1)dχ(M2d) ≥ 0. For Riemannian manifolds, the condition of non-positive sectional curvature is equivalent to being locally cat(0). The Hopf Conjecture subsequently has been generalized to include closed, piecewise Euclidean, locally cat(0) (generalized homology) manifolds. By work of M. W. Davis [D], Coxeter groups provide a rich source of piecewise Euclidean, locally cat(0) spaces. Given a flag triangulation of a (generalized homology) sphere Ln−1, a construction of Davis gives a reflection (generalized homology) orbifold O, with many (generalized homology) manifold covers. Recall that the f-polynomial fL of a simplicial complex L is defined by the formula fL(t) := ∑