Real tight contact structures on lens spaces and surface singularities

TL;DR

This paper classifies real tight contact structures on solid tori, applies these results to lens spaces and $S^3$, and constructs explicit examples using equivariant methods, revealing unique and bounded structures.

Contribution

It provides a classification of real tight contact structures on lens spaces and $S^3$, introducing explicit constructions and bounds for their counts.

Findings

Unique real tight structures on $S^3$ and $ ext{RP}^3$

At most one real tight $L(p, ext{±}1)$ with respect to one real structure

Existence of an invariant torus in $L(p,p-1)$ that cannot be made convex

Abstract

We classify 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 $S^3$ and real lens spaces $L(p,\pm 1)$. We prove that there is a unique real tight $S^3$ and $\mathbb{R}P^3$. We show there is at most one real tight $L(p,\pm 1)$ 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 $L(p,p-1)$ which cannot be made convex equivariantly.