Generation of sutured manifolds

TL;DR

This paper demonstrates that all connected sutured manifolds with a given surface boundary can be constructed from a basic product manifold using surgeries, with implications for Floer theories of 3-manifolds.

Contribution

It establishes a generation theorem for sutured manifolds with a fixed boundary surface, expanding understanding of their structure and applications in Floer homology.

Findings

Connected sutured manifolds are generated by product manifolds via surgery triads.

The result generalizes known folklore theorems for disks and spheres.

Applications in Floer theories of 3-manifolds are demonstrated.

Abstract

Given a compact, oriented, connected surface $F$, we show that the set of connected sutured manifolds $(M,\gamma)$ with $R_{\pm}(\gamma)\cong F$ is generated by the product sutured manifold $(F,\partial F)\times[0,1]$ through surgery triads. This result has applications in Floer theories of $3$--manifolds. The special case when $F=D^2$ or $S^2$ has been a folklore theorem, which has already been used by experts before.