Naturality in sutured monopole and instanton homology

TL;DR

This paper refines sutured monopole and instanton Floer homology invariants by assigning richer algebraic structures, enabling new functorial extensions and potential invariants for contact structures, knots, and bordered 3-manifolds.

Contribution

It introduces projectively transitive systems of modules as refined invariants, extending the functorial framework of sutured Floer theories.

Findings

Refined invariants assign richer algebraic objects to sutured manifolds.

Framework supports extension to contact structures, knots, and bordered 3-manifolds.

Lays foundation for future functorial and invariant constructions.

Abstract

Kronheimer and Mrowka defined invariants of balanced sutured manifolds using monopole and instanton Floer homology. Their invariants assign isomorphism classes of modules to balanced sutured manifolds. In this paper, we introduce refinements of these invariants which assign much richer algebraic objects called projectively transitive systems of modules to balanced sutured manifolds and isomorphisms of such systems to diffeomorphisms of balanced sutured manifolds. Our work provides the foundation for extending these sutured Floer theories to other interesting functorial frameworks as well, and can be used to construct new invariants of contact structures and (perhaps) of knots and bordered 3-manifolds.