G 2 -manifolds and associative submanifolds via semi-Fano 3 -folds

We construct many new topological types of compact G 2 -manifolds, that is, Riemannian 7 -manifolds with holonomy group G 2 . 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 G 2 -metrics. Many of the G 2 -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 G 2 -metrics. By varying the semi-Fanos used to build different G 2 -metrics on the same 7 -manifold we can change the number of rigid associative 3 -folds we produce.