Naturality of the contact invariant in monopole Floer homology under strong symplectic cobordisms

Echeverria, Mariano

doi:10.2140/agt.2020.20.1795

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Naturality of the contact invariant in monopole Floer homology under strong symplectic cobordisms

Mariano Echeverria

Algebraic & Geometric Topology 20 (2020) 1795–1875

DOI: 10.2140/agt.2020.20.1795

Abstract

The contact invariant is an element in the monopole Floer homology groups of an oriented closed three-manifold canonically associated to a given contact structure. A nonvanishing contact invariant implies that the original contact structure is tight, so understanding its behavior under symplectic cobordisms is of interest if one wants to further exploit this property.

By extending the gluing argument of Mrowka and Rollin to the case of a manifold with a cylindrical end, we will show that the contact invariant behaves naturally under a strong symplectic cobordism.

As quick applications of the naturality property, we give alternative proofs for the vanishing of the contact invariant in the case of an overtwisted contact structure, its nonvanishing in the case of strongly fillable contact structures and its vanishing in the reduced part of the monopole Floer homology group in the case of a planar contact structure. We also prove that a strong filling of a contact manifold which is an $L$ –space must be negative definite.

Keywords

contact invariant, monopole Floer homology

Mathematical Subject Classification 2010

Primary: 57R17, 57R58

References

Publication

Received: 7 September 2018

Revised: 11 August 2019

Accepted: 31 August 2019

Published: 20 July 2020

Authors

Mariano Echeverria
Department of Mathematics University of Virginia Charlottesville, VA United States	Department of Mathematics Rutgers University Piscataway, NJ United States

Volume 20, issue 4 (2020)