Families of four-dimensional manifolds that become mutually diffeomorphic after one stabilization (Q1868897)

In this paper, the author introduces a cut and paste move for four-manifolds, called a geometrically null log transform, and proves that any two manifolds related by a sequence of these moves become diffeomorphic after one stabilization (i.e., applying to the connected sum with the non-trivial \(S^{2}\)-bundle over \(S^{2}\)). The null log transform is introduced by using the symplectic fiber sum and the construction of Fintushel and Stern in [\textit{R. Fintushel} and \textit{R. Stern}, Invent. Math. 134, No. 2, 363-400 (1998; Zbl 0914.57015)]. Several families of \(4\)-manifolds are constructed by these methods and the members of any one of these families become diffeomorphic after one stabilization. This is proved by making use of Kirby calculus. Finally, the Seiberg-Witten invariants are computed to see that they are not diffeomorphic.