Stein-fillable open books of genus one that do not admit positive factorisations (Q6141578)

\textit{E. Giroux} [in: Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20--28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press; Singapore: World Scientific/distributor. 405--414 (2002; Zbl 1015.53049)] has established a one-to-one correspondence between isotopy classes of contact structures on a 3-manifold \(Y\) and positive stabilisation classes of open book decompositions of \(Y\), enabling one to consider questions of contact and symplectic geometry through a powerful lens of surface mapping class groups. In particular, a natural question when studying contact manifolds is that of fillability, i.e., determining when a contact manifold can be the boundary of a symplectic manifold in some compatible way. In the paper under review, the authors construct an infinite family of genus one open book decompositions supporting Stein-fillable contact structures and show that their monodromies do not admit positive factorisations. This extends a line of counterexamples in higher genera and establishes that a correspondence between Stein fillings and positive factorisations only exists for planar open book decompositions.