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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
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.

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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