Seiberg-Witten Floer Spectra and Contact Structures

TL;DR

This paper introduces a new homotopy-theoretic invariant of contact structures on rational homology 3-spheres using Seiberg-Witten Floer spectra, linking it to existing invariants and exploring its behavior under coverings.

Contribution

It defines a novel cohomotopy invariant of contact structures, establishes its relation to known Floer invariants, and introduces an equivariant version for finite covers.

Findings

The invariant recovers the classical contact element in Floer cohomology.

Properties of the invariant under finite coverings are analyzed.

A concrete example demonstrates the invariant's potential for new applications.

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 (2003). Furthermore, in light of the equivalence established in Lidman & Manolescu (2018a) between the Borel equivariant homology of said spectrum and the Seiberg-Witten Floer homology of Kronheimer & Mrowka (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, Mrowka, Ozsv\'ath & Szab\'o 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.