Does Kalai's $3^d$ conjecture hold for simplicial spheres?

Question

Kalai's $3^d$ conjecture asserts that every centrally symmetric $d$-polytope has at least $3^d$ non-empty faces. This is open in general, but has been proven for simplicial polytopes.

Question: Does the conjecture still hold for centrally symmetric simplicial spheres? Or are counterexamples known?

Here, a simplicial sphere is a simplicial complex that is homeomorphic to a sphere, but is not neccessarily the boundary complex of a simplicial polytope. I am not sure how to best formalize "centrally symmetric" in this context, but one idea would be to require a fixed-point free involution on the sphere.

Answer 1

The reference for Kalai's conjecture for simplicial polytopes is Richard Stanley's paper On the number of faces of centrally-symmetric simplicial polytopes which includes the theorem:

Let $A$ be a $(d - 1)$-dimensional Cohen-Macaulay simplicial complex, and suppose that a group $G$ of order $2$ acts freely on $A$. Then $$ h_i(A) \geq \binom{d}{i}$$ for $i$ from $0$ to $d$.

I think this theorem gives what you ask for.

Simplicial spheres (and all simplicial manifolds) are Cohen-Macauley by Reisner's theorem. The group $G$ acting freely is the same as being centrally-symmetric. And the relation between the $h_i$ and the number $f_i$ of faces of dimension $i$ gives

$$f_{i-1} = \sum_{k=0}^{i} \binom{d-k}{i-k} h_k \geq \sum_{k=0}^{i} \binom{d-k}{i-k} \binom{d}{k} = 2^i \binom{d}{i}$$

so $$ \sum_{i=0}^{d-1} f_i \geq \sum_{i=1}^d 2^i \binom{d}{i} = 3^d-1$$ which is Kalai's conjecture (after subtracting off the $d$-dimensional face which is counted as part of the simplicial polytope for Kalai's purposes but not counted as part of the simplicial sphere).