In a recent preprint, 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.

projectiveis crucial.) $\endgroup$