Seiberg-Witten Floer homotopy contact invariant

Abstract

We introduce a Floer homotopy version of the contact invariant introduced by Kronheimer-Mrowka-Ozváth-Szabó. 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.

53 pages, Theorem 1.5, 1.6, 1.7, 1.8, 1.9 are deleted, Theorem 1.5 and 1.6 have no new concrete examples, Theorem 1.7-1.9 contain some essential mistakes in the previous version