A monopole invariant for families of contact structures

TL;DR

This paper introduces a new invariant for families of contact structures on 3-manifolds using monopole Floer homology, enabling the detection of exotic loops and obstructions to certain fibrations.

Contribution

It generalizes the contact invariant to families, providing foundational results that connect this invariant with monopole Floer homology module structures.

Findings

Obstructs existence of sections in certain fibrations

Detects exotic loops of contact structures

Applies to links of surface singularities

Abstract

We use monopole Floer homology to study the topology of the space of contact structures on a 3-manifold. Our main tool is a generalisation of the Kronheimer--Mrowka--Ozsv\'ath--Szab\'o contact invariant to an invariant for families of contact structures, and we establish foundational results that describe the interaction between this invariant and the module structure in monopole Floer homology. We apply these results in several examples of contact manifolds, such as the links of non-rational surface singularities, to deduce several applications. Namely, we are able to obstruct the existence of sections of a natural fibration over the 2-sphere whose total space is the space of contact structures on the 3-manifold, and from this we are able to detect the existence of exotic loops of contact structures on contact 3-manifolds with convex sphere boundary.