Heegaard splittings and the tight Giroux Correspondence

This paper presents a new proof of the Giroux Correspondence for tight contact 3-manifolds using techniques from Heegaard splittings and convex surface theory. We introduce tight Heegaard splittings of arbitrary contact 3–manifolds; these generalise the Heegaard splittings naturally induced by an open book decomposition adapted to a contact structure on the underlying manifold. Via a process called refinement, any tight Heegaard splitting determines an open book, up to positive open book stabilisation. This allows us to translate moves relating distinct tight Heegaard splittings into moves relating their associated open books. We use this relationship to show that every Heegaard splitting of a contact 3-manifold may be stabilised to a Heegaard splitting induced by a supporting open book decomposition. Finally, wLzlky+/YAPZGnp+ZUWbUEfN2BNYqwe prove the tight Giroux Correspondence, showing that any pair of open book decompositions supporting a fixed tight contact structure become isotopic after a sequence of positive open book stabilisations and destabilisations.