Seiberg-Witten Floer homotopy contact invariant

TL;DR

This paper develops a Floer homotopy version of a contact invariant, establishes a gluing formula linking it to a Bauer-Furuta type invariant, and applies it to constrain symplectic fillings via equivariant KO-cohomology.

Contribution

It introduces a new Floer homotopy contact invariant and proves a gluing formula connecting it with existing invariants, enhancing understanding of contact structures and 4-manifold boundaries.

Findings

Established a Floer homotopy contact invariant.

Proved a gluing formula relating the invariant to Bauer-Furuta type invariant.

Provided constraints on symplectic fillings using equivariant KO-cohomology.

Abstract

We introduce a Floer homotopy version of the contact invariant introduced by Kronheimer-Mrowka-Ozv\'ath-Szab\'o. Moreover, we prove a gluing formula relating our invariant with the first author's Bauer-Furuta type invariant, which refines Kronheimer-Mrowka's invariant for 4-manifolds with contact boundary. As an application, we give a constraint for a certain class of symplectic fillings using equivariant KO-cohomology.