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?

Question

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 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.

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.

Answer 1

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. 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).

Answer 2

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 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<C,D> 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 $K$ is 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 conjectured by Nevo and Petersen that these parameters correspond to face numbers of a 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.

Answer 3

This page by Andrew Frohmader has these pieces of information:

• Stanley established the Charney-Davis conjecture for barycentric subdivisions of polytopes.
• "We [A.F.] show that it is enough to subdivide only the odd-dimensional faces, in addition to as many or as few of the even dimensional faces as one prefers."

Also, you might be interested in this note, which was just published this month:

"Even- vs. Odd-dimensional Charney–Davis Conjecture," Światosław R. Gal and Tadeusz Januszkiewicz, Discrete and Computational Geometry, Volume 44, No. 4, Dec. 2010. Here is the abstract:

Abstract 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.

Answer 4

A good reference explaining the motivation for the Charney-Davis conjecture is the article by Robin Forman in the Park City Geometric Combinatorics volume:

Geometric Combinatorics (Ias/Park City Mathematics Series) Editors: Ezra Miller, Victor Reiner, Bernd Sturmfels

At the end of his article is a discussion of recent work by P. Branden and S. Gal. Their papers should lead via mathscinet or google scholar to completely up-to-date references.