Summary: CORK TWISTING EXOTIC STEIN 4-MANIFOLDS
SELMAN AKBULUT AND KOUICHI YASUI
Abstract. From any 4-dimensional oriented handlebody X without 3- and
4-handles and with b2 1, we construct arbitrary many compact Stein 4-
manifolds which are mutually homeomorphic but not diffeomorphic to each
other, so that their topological invariants (their fundamental groups, homol-
ogy 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 - int Y is a handlebody without 3- and 4-handles and with b2 1, we con-
struct arbitrary many exotic embeddings of a compact 4-manifold Y into Z,
such that Y has the same topological invariants as Y .
A basic problem of 4-manifold topology is to find all exotic copies of smooth
4-manifolds, in particular to find various methods of constructing different smooth
structures on 4-manifolds (e.g. logarithmic transform , Fintushel-Stern's rational
blowdown  and knot surgery ). The purpose of this paper is to approach
this problem by corks and give applications. Since different smooth structures on a
4-manifold can be explained by existence corks which divide the manifold into two
Stein pieces , cork twisting Stein manifolds is a central theme of this paper.