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Symplectic instanton homology is an invariant for closed orientedthree-manifolds, defined by Manolescu and Woodward, which conjecturallycorresponds 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 $Y'$ betwo closed oriented three-manifolds, we show that the symplectic instantonhomology of their connected sum is isomorphic to the direct sum of the tensorproduct of their symplectic instanton homology, and a shift of their torsionproduct.  We define twisted versions of this homology, and then prove an analog of theFloer exact sequence, relating the invariants of a Dehn surgery triad. We usethis exact sequence to compute the rank of the groups associated to brancheddouble covers of quasi-alternating links, some plumbings of disc bundles overspheres, and some integral Dehn surgeries along certain knots.  We then define invariants for four dimensional cobordisms as maps between thesymplectic instanton homology of the two boundaries. We show that among thethree morphisms in the surgery exact sequence, two are such maps, associated tothe handle-attachment cobordisms. We also give a vanishing criteria for suchmaps associated to blow-ups.

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