Contact surgery distance

TL;DR

This paper introduces the contact surgery distance for contact 3-manifolds, establishing an upper bound relative to topological surgery distance, and classifies certain rational homology spheres based on invariants.

Contribution

It defines the contact surgery distance and proves it is at most five greater than the topological surgery distance, also classifies rational homology spheres with specific invariant properties.

Findings

Contact surgery distance is at most 5 larger than topological surgery distance.

Classification of rational homology 3-spheres based on $d_3$-invariant and $ ext{Gamma}$-invariant.

Establishment of bounds relating contact and topological surgery complexities.

Abstract

In this article, we define the contact surgery distance of two contact 3-manifolds $(M,\xi)$ and $(M',\xi')$ as the minimal number of contact surgeries needed to obtain $(M,\xi)$ from $(M',\xi')$. 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 $\Gamma$-invariant and Euler class.