Every real 3-manifold is real contact

TL;DR

This paper proves that every real 3-manifold admits a compatible real contact structure, extending contact topology to real manifolds and providing explicit constructions and classifications on lens spaces and $S^1\times S^2$.

Contribution

It demonstrates that all real 3-manifolds can be obtained via surgery from standard real $S^3$ and that this extends to contact structures, establishing the existence of real contact structures universally.

Findings

Every real 3-manifold admits a real contact structure.

Any overtwisted contact structure on a real homology 3-sphere can be made real.

Tight contact structures on lens spaces are real with respect to specific involutions.

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 real 3-manifold with a contact distribution that is antisymmetric with respect to the real structure. We show that every real 3-manifold can be obtained via surgery along invariant knots starting from the standard real $S^3$ and that this operation can be performed in the contact setting too. Using this result we prove that any real 3-manifold admits a real contact structure. As a corollary we show that any oriented overtwisted contact structure on an integer homology real 3-sphere can be isotoped to be real. Finally we give construction examples on $S^1\times S^2$ and lens spaces. For instance on every 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.