Contact surgery distance

In this article, we define the contact surgery distance of two contact 3-manifolds $(M,\xi )$ and $(M^{\mathrm{\prime }},{\xi }^{\mathrm{\prime }})$ as the minimal number of contact surgeries needed to obtain $(M,\xi )$ from $(M^{\mathrm{\prime }},{\xi }^{\mathrm{\prime }})$. Our main result states that the contact surgery distance between two contact $3$-manifolds is at most $5$ larger than the topological surgery distance between the underlying smooth manifolds. As a byproduct of our proof, we classify the rational homology $3$-spheres on which the $d_3$-invariant of a $2$-plane field already determines its $\mathrm{\Gamma }$-invariant and Euler class.