Stabilisation of 4-dimensional manifolds

The study of 4-manifolds leads one to an observation: pairs of exotic phenomena (referring to topologically identical yet smoothly distinct), disappear upon stabilisation with copies of S^2 x S^2. Wall shows that pairs of smooth, closed, simply-connected, homotopy equivalent 4-manifolds are h-cobordant and become diffeomorphic after sufficient stabilisation. Similarly, Baykur-Sunukjian show that pairs of embedded surfaces sharing the same homology class are smoothly isotopy after handle additions in the ambient 4-manifold, which under some additional \pi_1 condition simplifies to an ambient connected sum of the surface with S^1 x S^1. In this case it translates to a stabilisation of the corresponding double branched cover. In this talk I will present both proofs and discuss some additional examples of stabilisation results.