We explore the twisted connected sum (TCS) construction of compact G2 holonomy manifolds. A key ingredient in this construction is a pair of asymptotically cylindrical (ACYl) Calabi–Yau (CY) threefolds whose asymptotic K3 fibres satisfy suitable matching conditions. We first construct a complete dataset of projecting tops, which describe ACy CY threefolds. From these tops, we compute the associated building blocks and extract the lattice-theoretic data required for TCS constructions. We then formulate matching conditions for pairs of building blocks that guarantee the existence of a compatible hyper-Kähler rotation between their asymptotic K3 surfaces. Using these criteria, we perform a large-scale search for matching pairs within our dataset and compute the resulting topological invariants, including the Betti numbers, of the corresponding G2-manifolds.
The talk will introduce an accompanying open-source software package and database.