Common positive stabilisation of open book decompositions

The Giroux Correspondence states that two open book decompositions supporting the same contact structure are related by a sequence of positive open book stabilisations and destabilisations. In this note we show that any two open book decompositions supporting isotopic contact structures admit a common positive stabilisation.

💡 Research Summary

**
The paper refines a central result in three‑dimensional contact geometry, the Giroux correspondence, by showing that any two open book decompositions supporting isotopic contact structures admit a common positive stabilization. The classical Giroux correspondence asserts that two open books supporting the same contact structure can be related by a finite sequence of positive stabilizations and their inverses (positive destabilizations). However, this statement does not guarantee that the sequence can be arranged so that all stabilizations occur before any destabilizations. The authors fill this gap by proving that the mixed sequence can always be reordered into a block of stabilizations followed by a block of destabilizations, and consequently there exists a single open book that is a positive stabilization of both given open books.

The technical core of the argument is an elementary manipulation of the mapping‑class group of a surface. An abstract open book is a pair ((\Sigma,\phi)) where (\Sigma) is a compact surface with non‑empty boundary and (\phi) is a mapping class. A positive stabilization along a properly embedded arc (\alpha\subset\Sigma) consists of attaching a 1‑handle (H_\alpha) to (\Sigma) (so that the attaching region is (\partial\alpha\subset\partial\Sigma)), extending (\phi) by the identity over the new handle, and composing with a right‑handed Dehn twist (\tau_\alpha) about the simple closed curve formed by (\alpha) together with the core of the handle. Destabilization is the inverse operation.

The authors introduce a “complexity” function on open books, namely the negative Euler characteristic (-\chi(\Sigma)) of the page. Stabilization raises the complexity by one, destabilization lowers it by one. Given any sequence of stabilizations and destabilizations connecting ((\Sigma_1,\phi_1)) to ((\Sigma_n,\phi_n)), one can view the associated sequence of complexities as a walk on the integers. Local minima in this walk correspond to points where a destabilization is immediately followed by a stabilization. The key combinatorial step is to eliminate such minima.

Proposition 2.3 provides the necessary geometric move: if (\alpha) and (\beta) are two properly embedded arcs on a surface (\Sigma), then the open book obtained by stabilizing first along (\alpha) and then along (\beta), \