Corks, exotic 4-manifolds and genus functions

Abstract: We prove that every 4-dimensional oriented handlebody without 3- and 4-handles can be modified to admit infinitely many exotic smooth structures, and further that their genus functions are pairwise equivalent. We furthermore show that, for any 4-manifold admitting an embedding into a symplectic 4-manifold with weakly convex boundary, its genus function is algebraically realized as those of infinitely many pairwise exotic 4-manifolds. We also prove that, for any two 4-manifolds with non-degenerate intersection forms, algebraic inequivalences of genus functions are stable under connected sums and boundary sums with a certain type of 4-manifolds having arbitrarily large second Betti numbers. Furthermore, we introduce a notion of genus function type for diffeomorphism invariants and show that similar results hold for any such invariant. As an application of our exotic 4-manifolds, we also prove that for any (possibly non-orientable) 4-manifold, every submanifold of codimension less than two satisfying a mild condition can be modified to admit infinitely many exotically knotted copies.