Compatible Relative Lefschetz Fibrations On Admissible Relative Stein Pairs

For more than two decades it has been known that any compact Stein surface (of real dimension four) admits a compatible Lefschetz fibration over a two-disk. More recently, Giroux and Pardon have generalized this result by giving a complex geometric proof for the existence of compatible Lefschetz fibrations on Stein domains of any even dimension. As a preparatory step in proving the former, Akbulut and Ozbagci have shown that there exist infinitely many pairwise non-equivalent Lefschetz fibrations on the four-ball D^4 by using a result of Lyon constructing fibrations on the complements of (p, q)-torus links in the three-sphere. In this paper, we first extend this result to obtain compatible Lefschetz fibrations on D^6 whose pages are (p, q, 2)-Brieskorn varieties, and then construct a compatible “relative” Lefschetz fibrations on any Stein domain (of dimension six) which admit a certain (“admissible”) “relative Stein pair” structure. In particular, we provide a purely topological proof for the existence of Lefschetz fibrations on specific 6-dimensional Stein domains.