One stabilization is not enough for contractible 4-manifolds

Abstract

We construct an example of a cork that remains exotic after taking a connected sum with $S^2\times S^2$. Combined with a work of Akbulut-Ruberman, this implies the existence of an exotic pair of contractible 4-manifolds which remains absolutely exotic after taking a connected sum with $S^2\times S^2$.

22 pages, comments welcome! Major modification on arguments using bordered Floer homology