We construct many new topological types of compact -manifolds, that is, Riemannian -manifolds with holonomy group . 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 -folds built from semi-Fano -folds constructed previously by the authors. In many cases we determine the diffeomorphism type of the underlying smooth -manifolds completely; we find that many -connected -manifolds can be realized as twisted connected sums in a variety of ways, raising questions about the global structure of the moduli space of -metrics. Many of the -manifolds we construct contain compact rigid associative -folds, which play an important role in the higher-dimensional enumerative geometry (gauge theory/calibrated submanifolds) approach to defining deformation invariants of -metrics. By varying the semi-Fanos used to build different -metrics on the same -manifold we can change the number of rigid associative -folds we produce.
"-manifolds and associative submanifolds via semi-Fano-folds." Duke Math. J. 164 (10) 1971 - 2092, 15 July 2015. https://doi.org/10.1215/00127094-3120743