Surgery equivalence relations for 3-manifolds

TL;DR

This paper explores how restricted surgery operations define various equivalence relations among 3-manifolds, using mapping class groups and invariants like finite-type invariants to analyze their structure.

Contribution

It introduces a framework connecting surgery equivalence relations with filtrations of mapping class groups and discusses their characterization via 3-manifold invariants.

Findings

Different surgery restrictions lead to distinct non-trivial equivalence relations.

Mapping class group filtrations are central to understanding these relations.

Finite-type invariants help characterize and distinguish these equivalence classes.

Abstract

By classical results of Rochlin, Thom, Wallace and Lickorish, it is well-known that any two 3-manifolds (with diffeomorphic boundaries) are related one to the other by surgery operations. Yet, by restricting the type of the surgeries, one can define several families of non-trivial equivalence relations on the sets of (diffeomorphism classes of) 3-manifolds. In this expository paper, which is based on lectures given at the school ``Winter Braids XI'' (Dijon, December 2021), we explain how certain filtrations of mapping class groups of surfaces enter into the definitions and the mutual comparison of these surgery equivalence relations. We also survey the ways in which concrete invariants of 3-manifolds (such as finite-type invariants) can be used to characterize such relations.