Loops of handleslides for sutured diagrams

TL;DR

This paper establishes a new topological framework for sutured manifolds by analyzing the handle-slide complexes and defining invariants that extend uniquely, aiding in the naturality proofs of Floer homology theories.

Contribution

It introduces a handle-slide complex for sutured compression bodies and defines tight Heegaard invariants with unique extensions, advancing the understanding of Floer homology.

Findings

The handle-slide complex becomes simply connected after attaching specific 2-cells.

Each tight Heegaard invariant admits a unique extension to a strong invariant.

Provides a new approach for proving naturality in Floer homology theories.

Abstract

We prove that the cut-system complex of a sutured compression body, with vertices representing cut-systems and edges corresponding to handleslides, becomes simply connected when six kinds of 2-cells are attached. Moreover, we define tight Heegaard invariants and show that each admits a unique extension to a strong Heegaard invariant. This gives a new framework for proving naturality results for Floer homology theories associated to sutured manifolds.