Compact Stein surfaces as branched covers with same branch sets

Abstract

Loi and Piergallini showed that a smooth compact, connected $4$-manifold $X$ with boundary admits a Stein structure if and only if $X$ is a simple branched cover of a $4$-disk $D^4$ branched along a positive braided surface $S$ in a bidisk $D_{1}^{2} \times D_{2}^{2} \approx D^4$. For each integer $N \geq 2$, we construct a braided surface $S_{N}$ in $D^4$ and simple branched covers $X_{N, 1}, X_{N, 2}, \dots , X_{N, N}$ of $D^{4}$ branched along $S_{N}$ such that the covers have the same degrees, and they are mutually diffeomorphic, but the Stein structures associated to the covers are mutually not homotopic. Furthermore, by reinterpreting this result in terms of contact topology, for each integer $N \geq 2$, we also construct a transverse link $L_{N}$ in the standard contact $3$-sphere $(S^3, \xi_{std})$ and simple branched covers $M_{N,1}, M_{N,2}, \ldots, M_{N, N}$ of $S^3$ branched along $L_{N}$ such that the covers have the same degrees, and they are mutually diffeomorphic, but the contact structures associated to the covers are mutually not isotopic.