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This paper presents a new proof of the Giroux Correspondence for tightcontact $3$-manifolds using techniques from Heegaard splittings and convexsurface theory. We introduce tight Heegaard splittings, which generalise theHeegaard splittings naturally induced by an open book decomposition of acontact manifold. Via a process called refinement, any tight Heegaard splittingdetermines an open book, up to positive open book stabilisation. This allows usto translate moves relating distinct tight Heegaard splittings into movesrelating their associated open books. We use these tools to show that everyHeegaard splitting of a contact 3-manifold may be stabilised to a splittingassociated to a supporting open book decomposition. Finally, we prove the tightGiroux Correspondence, showing that any pair of open book decompositionscompatible with isotopic contact structures become isotopic after a sequence ofpositive open book stabilisations.

Heegaard splittings and the tight Giroux Correspondence