Homologie Instanton Symplectique : somme connexe, chirurgie enti\`ere, et applications induites par cobordismes

TL;DR

This thesis explores the properties of symplectic instanton homology for three-manifolds, establishing formulas for connected sums, Dehn surgeries, and cobordisms, and applying these to compute invariants for various topological constructions.

Contribution

It proves a K"unneth-type formula for connected sums, develops an exact sequence for Dehn surgeries, and defines cobordism maps, advancing the understanding of symplectic instanton homology.

Findings

Connected sum homology is isomorphic to a tensor product plus torsion shift.

An exact sequence relates invariants of Dehn surgery triads.

Cobordism maps are constructed and their vanishing criteria are identified.

Abstract

Symplectic instanton homology is an invariant for closed oriented three-manifolds, defined by Manolescu and Woodward, which conjecturally corresponds to a symplectic version of a variant of Floer's instanton homology. In this thesis we study the behaviour of this invariant under connected sum, Dehn surgery, and four-dimensional cobordisms. We prove a K\"unneth-type formula for the connected sum : let $Y$ and $Y'$ be two closed oriented three-manifolds, we show that the symplectic instanton homology of their connected sum is isomorphic to the direct sum of the tensor product of their symplectic instanton homology, and a shift of their torsion product. We define twisted versions of this homology, and then prove an analog of the Floer exact sequence, relating the invariants of a Dehn surgery triad. We use this exact sequence to compute the rank of the groups associated to branched double covers of quasi-alternating links, some plumbings of disc bundles over spheres, and some integral Dehn surgeries along certain knots. We then define invariants for four dimensional cobordisms as maps between the symplectic instanton homology of the two boundaries. We show that among the three morphisms in the surgery exact sequence, two are such maps, associated to the handle-attachment cobordisms. We also give a vanishing criteria for such maps associated to blow-ups.