Matt also asked: "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?"

So let me add (gradually, helped by Eran Nevo) more information regarding the Charney-Davis conjecture and its analogs and extensions.

8) **The CD-index, order complexes of regular CW spheres and Gorenstein* spaces.**

Starting with a poset P you can define a simplicial complex, called the chain complex of P, whose faces are chains in P. Chain-complexes of a posets are always flag. Chain-complexes of *graded* posets have the property of being "completely balanced" or "colored". If you color each element by its grade, every face is properly colored (it contains no two vertices of the same color). If P is a graded poset and all maximum chains have d elements then its chain complex is (d-1)-dimensional completely balanced simplicial complex.

Especially important are posets of faces of pure *regular CW-complexes*. (Here we will color an i-face by color 'i'.) We define now in a few steps the CD-index. Let P be a regular CW-complex.

i) For a subset S of indices we let $f_S(P)$ be the number of chains of faces whose dimensions are prescribed by S. (Equivalently, the number of (|S|-1)-dimensional faces of the chain complex whose vertices are collored by the indices of S.)

ii) Define $\beta_S(P)$ (Also denoted by $h_S(P)$) by
$$\beta_S(P)=\sum _{R \subset S}(-1)^{|S\backslash R|}f_T(P).$$

(This definition extend to completely balanced pure simplicial complexes.) Consider the generating function in two noncommuting variables 'a' and 'b' describing the $2^d$ parameters $\beta_S(P)$ as follows. $$\beta(P)=\sum \beta_S(P)w_S.$$
where $w_S$ is a word in 'a's and 'b's where you put in the ith place an 'a' if i belongs to S and an 'a' otherwise.

iii) If P is Eulerian than the polynomial $\beta_S(P)$ is symmetric w.r.t. replacing 'a' and 'b'.
(This was proved by Stanley and it extends to Eulerian completely balanced simplicial complexes.)

iv) Bayer and Billera showed that for Eulerian regular CW-complexes (and, more generally, graded Eulerian posts) the linear space spanned by the $f_S$ vectors (or $b_S$ polynomials$) has dimension which is the dth Fibonacci number.

v) For Eulerian graded posets Jonathan Fine defined remarkable Fibonacci number of parameters: the coefficients of a certain degree d polynomial Q in two noncommuting variables C (of degree 1) and D (of degree 2). Fine showed that the $beta(P)$ polynomial can be expressed as a polynomial in the two non-commuting variables C=a+b and D=ab+ba!

vi) The coefficient of $D^{d/2}$ is precisely the Charney-Davis expression.

vii) Kalle Karu proved that for face postes of regular CW spheres of dimension (d-1) (and more generally Gorensetein* regular CW spheres) these coefficients are nonnegative. This supplies important special cases for the Charney Davis conj.

**Some link:** A blog post on Bayer-Billera's theorem and the CD index; Bayer and Klapper's paper on the CD index' and list of papers citing it (CiteseerX); Karu's paper;

9) **Gal's $\gamma$-numbers.**

Let $K$ be a simplicial $(d-1)$-dimensional simplicial complex. The $h$-vector of $K$, defined by

$$ \sum_{0\leq i\leq d}h_i(K)x^{d-i}= \sum_{0\leq i\leq d}f_{i-1}(K)(x-1)^{d-i}.$$

If $K$ is Eulerian then we have the Dehn-Sommerville relations: $$h_i=h_{d-i}$$.

The polynomial $$h(K,x)=\sum h_i(K)x^{d-i}$$ can be expressed (uniquely) as a linear combination of monomials $x^i(1+x)^{d-2i}$.

$$h(K,x)= \sum \gamma_i(K)x^i(1+x)^{d-2i}.$$

**Gal Conjecture:** If K is a flag Gorenstein* (d-1)-dimensional complex (In particular, a triangulation of the (d-1)-sphere) then $\gamma_i(K) \ge 0$.

When $d$ is even, Gal's top coefficient is the Charney-Davis parameter.

Moreover it is conjecture by Nevo and Petersen that these parameters correspond to face number of completely balanced simplicial complex.

It was also conjectured that the $h$-polynomial for flag triangulated spheres has real zeroes. This strong form of Charney-Davis conjecture was disproved by Światosław Gal as well.

Links: Gal's paper; the paper by Nevo and Petersen.