Cork twisting exotic Stein 4-manifolds

Abstract

From any 4-dimensional oriented handlebody $X$ without 3- and 4-handles and with $b_2 \ge 1$, we construct arbitrary many compact Stein 4-manifolds that are mutually homeomorphic but not diffeomorphic to each other, so that their topological invariants (their fundamental groups, homology groups, boundary homology groups, and intersection forms) coincide with those of $X$. We also discuss the induced contact structures on their boundaries. Furthermore, for any smooth 4-manifold pair $(Z, Y)$ such that the complement $Z − \operatorname{int} Y$ is a handlebody without 3- and 4-handles and with $b_2 \ge 1$, we construct arbitrary many exotic embeddings of a compact 4-manifold $Y'$ into $Z$, such that $Y'$ has the same topological invariants as $Y$.