TOPOLOGY OF MULTIPLE LOG TRANSFORMS OF 4-MANIFOLDS

Abstract

Given a 4-manifold X and an imbedding T2 × B2 ⊂ X, we describe an algorithm X ↦ Xp,q for drawing the handlebody of the 4-manifold obtained from X by (p, q)-logarithmic transforms along the parallel tori. By using this algorithm for relatively prime (p, q), we obtain a simple handle picture of the Dolgachev surface E(1)p,q, from that we deduce that the exotic copy [Formula: see text] of [Formula: see text] differs from the original one by a codimension zero simply connected Stein submanifold Mp,q. This gives examples of infinitely many small Stein manifolds which are exotic copies of each other (rel boundaries). Also by using the description of S2 × S2 as a union of two cusps glued along their boundaries, and by using this algorithm, we show that a pair of log transforms along the tori in these cusps gives back S2 × S2 or [Formula: see text].