The Weinstein conjecture for connected sums (Q2797796)

In dimension 3, \textit{H. Hofer} [Invent. Math. 114, No. 3, 515--563 (1993; Zbl 0797.58023)] proved that if a closed connected contact manifold \((M,\ker(\alpha))\) can be written as a contact connected sum of two contact manifolds \((M_1,\xi_1)\) and \((M_2,\xi_2)\), then either (i) there is a contractible Reeb orbit, or (ii) one of the \((M_i,\xi_i)\) is contactomorphic to the standard tight contact \((S^3,\xi_{st})\).NEWLINENEWLINEIn the paper under review, the authors generalize this result to higher dimensions under some conditions on \(\pi_1(M)\). One key ingredient is to show that the evaluation map from the boundary of the moduli of pseudo-holomorphic disks (with appropriate Lagrangian boundary conditions) to the belt sphere of the handle is a degree-one map. Some technical details of this result are obtained in [Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. Indice della Serie IV (dal 1, 1974 al 18, 1990). Pisa: Scuola Normale Superiore (1991; Zbl 0743.00010)]. After that, a deformation trick is applied to the evaluation map to extract more precise topological information about the map to obtain the final result.