Topology of multiple log transforms of 4-manifolds

TL;DR

This paper develops an algorithm to visualize and analyze the effects of multiple log transforms on 4-manifolds, revealing new exotic Stein structures and demonstrating invariance of certain manifolds under these transforms.

Contribution

It introduces a novel algorithm for handlebody diagrams of log transforms, leading to new examples of exotic Stein manifolds and insights into their smooth structures.

Findings

Constructed handlebody pictures of Dolgachev surfaces E(1)_{p,q}

Produced infinitely many exotic Stein submanifolds with same boundary

Showed log transforms do not alter the smooth structure of S^2 x S^2

Abstract

Given a 4-manifold X and an imbedding of T^{2} x B^2 into X, we describe an algorithm X --> X_{p,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, we obtain a simple handle picture of the Dolgachev surface E(1)_{p,q}, from that we deduce that the exotic copy E(1)_{p,q} # 5(-CP^2) of E(1) # 5(-CP^2) differs from the original one by a codimension zero simply connected Stein submanifold M_{p,q}, which are therefore examples of infinitely many Stein manifolds that are exotic copies of each other (rel boundaries). Furthermore, by a similar method we produce infinitely many simply connected Stein submanifolds Z_{p} of E(1)_{p,2} # 2(-CP^2)$ with the same boundary and the second Betti number 2, which are (absolutely) exotic copies of each other; this provides an alternative proof of a recent theorem of the author and Yasui [AY4]. Also, by using the description of S^2 x S^2 as a union of two cusps glued along their boundaries, and by using this algorithm, we show that multiple log transforms along the tori in these cusps do not change smooth structure of S^2 x S^2.