#### Volume 18, issue 3 (2018)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Compact Stein surfaces as branched covers with same branch sets

### Takahiro Oba

Algebraic & Geometric Topology 18 (2018) 1733–1751
##### Abstract

For each integer $N\ge 2$, we construct a braided surface $S\left(N\right)$ in ${D}^{4}$ and simple branched covers of ${D}^{4}$ branched along $S\left(N\right)$ such that the covers have the same degrees and are mutually diffeomorphic, but Stein structures associated to the covers are mutually not homotopic. As a corollary, for each integer $N\ge 2$, we also construct a transverse link $L\left(N\right)$ in the standard contact $3$–sphere and simple branched covers of ${S}^{3}$ branched along $L\left(N\right)$ such that the covers have the same degrees and are mutually diffeomorphic, but contact manifolds associated to the covers are mutually not contactomorphic.

##### Keywords
compact Stein surfaces, branched coverings, Lefschetz fibrations, contact manifolds
##### Mathematical Subject Classification 2010
Primary: 57M12, 57R17
Secondary: 32Q28, 57R65