Compact Stein surfaces as branched covers with same branch sets

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.