G₂-Manifolds and associative submanifolds via semi-fano 3-folds

We construct many new topological types of compact G2-manifolds, that is, Riemannian 7-manifolds with holonomy group G₂. To achieve this we extend the twisted connected sum construction first developed by Kovalev and apply it to the large class of asymptotically cylindrical Calabi-Yau 3-folds built from semi-Fano 3-folds constructed previously by the authors. In many cases we determine the diffeomorphism type of the underlying smooth 7-manifolds completely; we find that many 2-connected 7-manifolds can be realized as twisted connected sums in a variety of ways, raising questions about the global structure of the moduli space of G₂-metrics. Many of the G₂-manifolds we construct contain compact rigid associative 3-folds, which play an important role in the higher-dimensional enumerative geometry (gauge theory/ calibrated submanifolds) approach to defining deformation invariants of G₂-metrics. By varying the semi-Fanos used to build different G₂-metrics on the same 7-manifold we can change the number of rigid associative 3-folds we produce.