Contact surgery distance

Abstract

In this article, we define the contact surgery distance of two contact 3-manifolds $(M,\hat{I}¾)$ and $(M^{\prime },\hat{I}{¾}^{\prime })$ as the minimal number of contact surgeries needed to obtain $(M,\hat{I}¾)$ from $(M^{\prime },\hat{I}{¾}^{\prime })$. 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 $\hat{I}“$-invariant and Euler class.

24 pages, 3 pictures. Comments welcome!