In a [recent preprint](https://arxiv.org/abs/1812.10454), Adiprasito proves that if $\Delta$ is a simplicial complex of dimension $d$ that can be embdedded in a $2d$-dimensional homology sphere (say $\Sigma$) that satisfies *a version of the hard Lefschetz theorem*, then:
$$ f\_{d}(\Delta) \leq (d+2)f\_{d-1}(\Delta),$$
where $f\_{k}(\Delta)$ is the number of faces of dimension $k$ of $\Delta$. This can be seen as higher dimensional analogue of Euler's inequality for a simple connected planar graph $\mathcal{C}$:


$$f\_{1}(\mathcal{C}) \leq 3f\_{0}(\mathcal{C})-6.$$


As far as I understand, the equality proved by Adiprasito is a version of the hard Lefschetz theorem for *an analogue* of the simplicial homology of the pair $(\Sigma, \Delta)$. Indeed, denote by $A\_{\bullet}(X, \mathbb{Z})$ this analogue of the simplicial homology of a simplicial space.


Adiprasito explains in his preprint that we have the inequalities (or at least this what I have understood, I might be mistaken)


\_$\dim A\_{d}(\Delta, \mathbb{Z}) \leq f\_{d-1}(\Delta)$, because the $d$-chains are $(d-1)$-dimensional faces.


\_$\dim A\_{d+1}(\Delta, \mathbb{Z}) \geq f\_{d}(\Delta) - (d+1)f\_{d-1}(\Delta)$, because $(d+1)$-chains are $d$-dimensional faces and (by construction of the simplicial complex?) each $(d-1)$-dimensional face gives exactly $d+1$ relations.


Now, assume there is an isomorphism $l : A\_{d+1}(\Sigma, \mathbb{Z}) \longrightarrow A\_{d}(\Sigma, \mathbb{Z})$, which respects $A\_{\bullet}(\Delta, \mathbb{Z})$. If the map $A\_{d+1}(\Delta, \mathbb{Z}) \longrightarrow A\_{d+1}(\Sigma, \mathbb{Z})$ is injective, then, a simple diagram chasing, shows that $l$ induces an injection:
$$ l : A\_{d+1}(\Delta, \mathbb{Z}) \longrightarrow A\_{d}(\Delta, \mathbb{Z}).$$
And we get the desired inequality.


I have a a few questions about this very nice argument:


\_I have some troubles to understand concretly what is $A(\Delta)\_{\bullet}$. Any suggestions?


\_Adiprasito goes on proving that there are some (many?) PL-spheres that do satisfy the hard Lefschetz theorem by construction some explicit Lefschetz maps. I must say that this construction is quite involved for me and I am not able to follow it. Are there some simple examples of spheres (even for the standard one, I will be happy to understand) for which the Lefschetz map can be constructed explicitely (and relatively easily)?


\_Is there a general argument to prove that the map $A\_{d+1}(\Delta, \mathbb{Z}) \longrightarrow A\_{d+1}(\Sigma, \mathbb{Z})$ is injective? I might be totally mistaken on this last point at is it seems to be quite the opposite of the weak Lefschetz theorem in algebraic geometry.



Inequality number of facets simplicial complex
I follow your notation rather than mine.


$A^\bullet(\Delta)$ is obtained as follows: You construct a linear system of parameters for $\Sigma$. If $\Sigma$ is of dimension $d-1$, then this is of length $d$. In fact, you can think of this linear system as a set of coordiantes for the vertices of $\Sigma$ in $\mathbb{R}^d$. You restrict that linear system to the face ring for $\Delta$. Note that this is in general MORE that the linear system of parameters for $\Delta$: If $\Delta$ is of dimension $k-1$, you have $d-k$ linear forms to many. In general, I think of simplicial complexes as coming with some coordinates, making this well-defined.


In particular, the surjection $$A^\bullet(\Sigma) \rightarrow A^\bullet(\Delta)$$ (or dually the injection you write down) just follows from definition.


As for constructing Lefschetz maps, Sam is right for the classical case of projective toric varieties. My construction of Lefschetz elements for general simplicial spheres is more involved, but the case of vertex decomposable spheres is perhaps understandable in an easier way. 


