Every real $3$-manifold is real contact

Abstract

A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution which is called a real structure. A real contact 3-manifold is a 3-manifold with a contact distribution which is mapped to itself by the real structure in an orientation reversing manner. We show that every real 3-manifold can be obtained via surgery along invariant knots starting from the standard real 3-sphere. Moreover we show that this operation can be performed in the contact setting. Using this result we show that any real 3-manifold admits a real contact structure. On any lens space there exists a unique real structure that acts on each Heegaard torus as hyperellipic involution. We show that any tight contact structure on any lens space is real with respect to that real structure.