Associative Submanifolds in Twisted Connected Sum G₂-Manifolds

We introduce a method to construct closed rigid associative submanifolds in twisted connected sum (Formula presented) -manifolds. More precisely, we prove a gluing theorem of asymptotically cylindrical (ACyl) associative submanifolds in ACyl (Formula presented) -manifolds under a transverse intersection hypothesis. This is analogous to the gluing theorem for (Formula presented) -instantons introduced in [29]. We rephrase the hypothesis in the special cases where the ACyl associative submanifolds are obtained from holomorphic curves or special Lagrangians in ACyl Calabi-Yau (Formula presented) -folds, in terms of algebraic-geometric conditions and topological conditions, respectively. This yields many rigid associative submanifolds with new topological types (Formula presented), (Formula presented) or (Formula presented) .