Diffeomorphism classes of the doubling Calabi–Yau threefolds with Picard number two

Previously we constructed Calabi–Yau threefolds by a differential-geometric gluing method using Fano threefolds with their smooth anticanonical K 3 divisors (Bini and Iacono in Rend Semin Mat Univ Politec Torino 73(1):9–20, 2015). In this paper, we further consider the diffeomorphism classes of the resulting Calabi–Yau threefolds (which are called the doubling Calabi–Yau threefolds ) starting from different pairs of Fano threefolds with Picard number one. Using the classifications of simply-connected 6-manifolds in differential topology and the λ - invariant introduced by Lee (Internat J Math 21:701–725, 2010), we prove that any two of the doubling Calabi–Yau threefolds with Picard number two are not diffeomorphic to each other when the underlying Fano threefolds are distinct families.