Hard Lefschetz property for 𝕊³-actions
José Ignacio Royo Prieto, Martintxo Saralegi-Aranguren, Robert A. Wolak · Proceedings of the American Mathematical Society · 2024
The Hard Lefschetz Property (HLP) has recently been formulated in the context of isometric flows without singularities on manifolds. In this category, two versions of the HLP (manifold and transverse) have been proven to be equivalent, thus generalizing the results in the important classes of K-contact and Sasakian manifolds. In this paper we define both versions of the HLP for almost-free S 3 \mathbb {S}^3 -actions, and prove that they agree for actions satisfying a cohomological condition, which holds in the important category of 3-Sasakian manifolds, where those two versions of the HLP are shown to be true. We also provide a family of examples of free actions of the 3-sphere which are not 3-Sasakian manifolds, but satisfy the HLP.