Non-existence theorems on infinite order corks

Abstract

Suppose that X,X′ are simply-connected closed exotic 4-manifolds. It is well-known that X′ is obtained by an order 2 cork twist of X. We give an infinite family of exotic 4-manifolds not generated by any infinite order cork. This is the first example admitting such a condition. We prove a necessary condition of 4-dimensional Ozsváth-Szabó invariants for a family to be generated by an infinite order cork and as an application give non-contractible relatively exotic 4-manifolds that are never induced by any cork. Furthermore, we give an estimate of the number of Ozsváth-Szabó invariants of 4-manifolds generated by a cork.