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

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.

• Your definition of centrally symmetric simplicial complex is the standard one: see, e.g., arxiv.org/abs/1711.09310 Sep 7 at 13:59

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 $$d$$.
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).