Every real $3$-manifold is real contact

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 $3$ 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 over-twisted 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.

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Received 7 October 2021

Accepted 6 February 2022

Published 26 April 2023