What I would like to know is exactly what the title asks:



> 
> What are the most general classes of
>  simplicial complexes or posets for
>  which the Charney-Davis conjecture is
>  known, and what is the most general
>  setting for which it might expected to be
>  true?
> 
> 
> 


I believe it conjectured, for example, that for a [flag](http://en.wikipedia.org/wiki/Clique_complex) simplicial sphere (i.e. a flag simplicial complex which is homeomorphic to a sphere) of dimension $2d-1$, 
$(-1)^d (1- \frac{1}{2}f\_0 + \frac{1}{4} f\_1 - \frac{1}{8} f\_2 + \dots + (\frac{1}{2})^{2d} f\_{2d-1} ) \ge 0$, but that this is still not known. (Here $f\_i$ is the number of $i$-dimensional faces.) 


I am guessing that the case of simplicial polytopes follows easily from Stanley's $g$-theorem, but what are the most general classes of spheres for which this statement is known? **This guess is not correct** see [this asnwer](https://mathoverflow.net/questions/50258/what-are-the-most-general-classes-of-simplicial-complexes-or-posets-for-which-th/92976#92976).


I am aware that there are analogous statements expected to be true for other triangulated manifolds besides spheres, perhaps for more general kinds of posets, etc. But what exactly are these conjectural statements in their most general forms?


Any pointers to survey articles would be greatly appreciated.



 What are the most general classes of simplicial complexes or posets for which the Charney-Davis conjecture is known, and what is the most general setting for which it might expected to be true?
There is quite a lot that can be said about the Charney-Davis conjecture, so let me say a few things and perhaps add more later.


1) The conjecture is a discrete analogue of a well known conjecture by Hopf on the Euler characteristic of nonpositively curved manifolds in odd dimension. You consider cubical complexes and the condition of nonpositive curvature is replaced (the CAT version) by the condition that all links of vertices (which are simplicial spheres to start with) are flag. (Namely they are the clique complexes of their 1-skeletons.)


2) The original formulation is in terms of the h-vectors and it says that if K is a flag complex which is a sphere of dimension d-1, and d=2e, then 



> 
> $(-1)^e(h\_0-h\_1+\dots+h\_d)\ge 0$.
> 
> 
> 


I suppose that this is equivalent to Matt's formulation in terms of the face number $f\_i$. ($f\_i(K)$ is the number of i-dimensional faces of K.) 


3) The conjecture as is, should extend to the case where K is a Gorenstein\* complex (i.e. a homology sphere in the weakest possible sense with coefficients in a field k).


4) The conjecture **is not known** to hold for th case where K is the boundary complex of a simplicial polytope! 


5) Stanley's book on combinatoris and commutative algebra (second edition p.100) is a good source (but a lot have happened later).


Let me mention two remarkable case where the conjecture is proven.


6) For 3-dimensional flag spheres. This is an extremely difficult result by Davis and Okun.: Michael Davis and Boris Okun, Vanishing theorems and conjectures for the $\ell^2$-homology of right-angled Coxeter groups. Geom. Topol. 5 (2001), 7–74. Archive version [link text](http://arxiv.org/abs/math.GR/0102104). The proof relies on a 1995 deep results by 
J Lott and W L¨uck, $L^2$ –topological invariants of 3–manifolds, Invent. Math. 120 
(1995) 15–60.


For 3-dimensional flag triangulations of spheres the Charney-Davis conjecture asserts that $f\_1 \ge 5f\_0 -15$. Barnett's lower bound conjecture asserts that for every triangulation of 3-dimensional sphere $f\_1 \ge 4f\_0-10$ and it is known that this statement extends to triangulated manifolds and pseudomanifolds. Likewise perhaps the 3-dimensional Charney David conjecture, namely Davis-Okun's theorem, extends to arbitrary flag-triangulated pseudomanifolds. 


7) For certain polytopes described by a geometric condition on their fans which is stronger than being flag. This is a remarkable result by N.C. Leung and V. Reiner, "The signature of a toric variety", Duke J. Math., 111(2002), 253-286. Preprint ([ps file](http://www.math.umn.edu/~reiner/Papers/Signature.ps)).


