Contact open books with exotic pages (Q2350502)

If a Stein filling \(X_a\) of a contact 3-manifold \(X\) is homeomorphic to any Stein filling \(X_b\) but not diffeomorphic, then \(X_a\) is called exotic. \textit{S. Akbulut} and \textit{K. Yasui} [J. Symplectic Geom. 12, No. 4, 673--684 (2014; Zbl 1322.32008)] gave an example of a closed 3-manifold \(\partial X\) that is the boundary of a 4-manifold \(X\) carrying two contact structures \(\eta\) and \(\eta'\) such that \((\partial X,\eta)\) and \((\partial X,\eta')\) admit infinite families \(\{X_p;\;p\;\text{odd}\}\) and \(\{X_p;\;p\;\text{even}\}\), respectively, of pairwise exotic simply-connected Stein fillings with \(b_2=2\). Let \(M_p=\text{OB}(X_p,\text{id})\) be a closed \(5\)-manifold viewed as the open book with page \(X_p\) and monodromy the identity map \(\text{id}\). For each \(p\), \(X_p\) admits a Stein structure and \(\text{id}\) is a symplectomorphism, and there is a contact structure \(\xi_p\) on \(M_p\) supported by this open book. In this paper, the authors consider a fixed contact \(3\)-manifold that admits infinitely many compact Stein fillings which are all homeomorphic but pairwise non-diffeomorphic. They prove that if \(p\) is odd, then \(M_p\) is diffeomorphic to \(S^2\times S^3\#S^2\times S^3\), and if \(p\) is even, then \(M_p\) is diffeomorphic to \(S^2\times S^3\#S^2\widetilde\times S^3\), where \(S^2\widetilde\times S^3\) is the non-trivial \(S^3\)-bundle over \(S^2\). Also, they show that \((M_p,\xi_p)\) is contactomorphic to \((M_p,\xi_{p'})\) if and only if \(p=p'\).