On the symplectic fillings of standard real projective spaces

TL;DR

This paper investigates the symplectic fillability of standard contact structures on real projective spaces, proving non-fillability in certain cases, establishing simple connectivity of fillings, and providing a new proof of a classical theorem on sphere fillings.

Contribution

It offers a geometric proof that standard contact structures on real projective spaces are not Liouville fillable for odd dimensions greater than or equal to 3, and presents a new proof of a key theorem on sphere fillings.

Findings

Standard contact structures on real projective spaces are not Liouville fillable for odd dimensions ≥ 3.

All semipositive fillings of these contact structures are simply connected.

A new proof of the Eliashberg-Floer-McDuff theorem on symplectically aspherical fillings of spheres.

Abstract

We prove, in a geometric way, that the standard contact structure on the real projective space of dimension $2n-1$ is not Liouville fillable for $n \ge 3$ and odd. We also prove that, for all $n$, semipositive fillings of those contact structures are simply connected. Finally we give yet another proof of the Eliashberg-Floer-McDuff theorem on the diffeomorphism type of the symplectically aspherical fillings of the standard contact structure on the $(2n-1)$-dimensional sphere.