The algebraic $g$-conjecture for rational homology spheres

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The algebraic $g$-conjecture for rational homology spheres

   Let $\Delta$ be a rational homology sphere. It has the strong Lefschetz property if there exist a linear system of parameters $\Theta$ for $\mathbb{Q}[\Delta]$ and a linear form $\omega\in\mathbb{Q}[\Delta]$ such that multiplication by $\omega^{d-2i}$ induces an isomorphism

   $$\cdot\omega^{d-2i}: (\mathbb{Q}[\Delta]/\Theta)_i\longrightarrow (\mathbb{Q}[\Delta]/\Theta)_{d-i}$$

   for every $i\leq d/2$. The algebraic $g$-conjecture. Every rational homology sphere has the strong Lefschetz property. The source reports that Adiprasito announced a thorough solution of this conjecture, so it is recorded as solved.

   source: Feifei Fan, “Weak Lefschetz property of PL-spheres”, arXiv:2001.06594 (2021).