We prove the Weak Lefschetz theorem for simplicial PL-spheres. This result is weaker than the Hard Lefschetz theorem for more general spheres proved by Adiprasito (arXiv:1812.10454), but the proof here involves simple algebra and avoids the more complicated combinatorics.
The Hard Lefschetz Theorem for PL spheres
FAQ on the g-theorem and the hard Lefschetz theorem for face rings
Ehrhart Theory for Lawrence Polytopes and Orbifold Cohomology of Hypertoric Varieties
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