Topology of multiple log transforms of 4-manifolds
Selman Akbulut

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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
