Seiberg–Witten Floer spectra and contact structures

Abstract

In this article, the author defines an invariant of rational homology 3-spheres equipped with a contact structure as an element of a cohomotopy set of the Seiberg–Witten Floer spectrum as defined in Manolescu (Geometry Topol 7(2):889–932, 2003). Furthermore, in light of the equivalence established in Lidman and Manolescu (Astérisque 399:25, 2018) between the Borel equivariant homology of said spectrum and the Seiberg–Witten Floer homology of Kronheimer and Mrowka (Monopoles and three-manifolds, vol. 10, Cambridge University Press, Cambridge, 2007), the author shall show that this homotopy theoretic invariant recovers the already well known contact element in the Seiberg–Witten Floer cohomology (vid. e.g. Kronheimer et al. in Ann Math 20:457–546, 2007) in a natural fashion. Next, the behaviour of the cohomotopy invariant is considered in the presence of a finite covering. This setting naturally asks for the use of Borel cohomology equivariant with respect to the group of deck transformations. Hence, a new equivariant contact invariant is defined and its properties studied. The invariant is then computed in one concrete example, wherein the author demonstrates that it opens the possibility of considering scenarios hitherto inaccessible.