Real tight contact structures on lens spaces and surface singularities

Abstract

We give a partial classification for the real tight contact structures on solid tori up to equivariant contact isotopy and apply the results to the classification of real tight structures on [Formula: see text] and real lens spaces [Formula: see text]. We prove that there is a unique real tight [Formula: see text] and [Formula: see text]. We show that there is at most one real tight [Formula: see text] with respect to one of its two possible real structures. With respect to the other we give lower and upper bounds for the count. To establish lower bounds we explicitly construct real tight manifolds through equivariant contact surgery, real open book decompositions and isolated real algebraic surface singularities. As a by-product we observe the existence of an invariant torus in an [Formula: see text] which cannot be made convex equivariantly.