Fillings of Genus–1 Open Books and 4–Braids

Abstract

Abstract. We show that there are contact 3-manifolds of support genus one which admit infinitely many Stein fillings, but do not admit arbitrarily large ones. These Stein fillings arise from genus-1 allowable Lefschetz fibrations with distinct homology groups, all filling a fixed minimal genus open book supporting the boundary contact 3-manifold. In contrast, we observe that there are only finitely many possibilities for the homology groups of Stein fillings of a given contact 3-manifold with support genus zero. We also show that there are 4-strand braids which admit infinitely many distinct Hurwitz classes of quasipositive factorizations, yielding in particular an infinite family of knotted complex analytic annuli in the 4-ball bounding the same transverse link up to transverse isotopy. These realize the smallest possible examples in terms of the number of boundary components a genus-1 mapping class and the number of strands a braid can have with infinitely many positive/quasipositive factorizations.

Corks, covers, and complex curves

“…The literature already contains a variety of inequivalent monodromy factorizations giving rise to Lefschetz fibrations whose total spaces are homeomorphic but not diffeomorphic. In contrast, previous examples of braided surfaces arising from inequivalent braid factorizations (e.g., [Rud83a,Aur15,BVHM18,Oba20]) are not topologically equivalent, as they are distinguished by the fundamental groups of their complements or the homotopy types of their branched covers. The braid factorizations in Theorem D give the first examples in which the corresponding braided surfaces are exotically knotted.…”

“…Rudolph showed that such a surface can be realized as a compact piece of an algebraic curve. Crucially, this surface depends not only on the braid but on the quasipositive factorization itself, allowing for subtle control of the resulting surface (c.f., [BKW10,BVHM18,Hay19,Oba20]). We show that the disks from Figure 1 arise from "exotic" braid factorizations.…”

Higher-dimensional contact manifolds with infinitely many Stein fillings

“…Baykur and Van Horn-Morris [5] recently constructed infinitely many 4-braids each of which admits infinitely many inequivalent quasipositive factorizations. Their construction is based on the open book adapted to the standard contact 3-torus constructed by Van Horn-Morris [32] whose page is diffeomorphic to Σ 1,3 and whose monodromy is…”

“…This enables us to study Stein fillings of a given contact 3-manifold by Lefschetz fibrations. For example, by using Lefschetz fibrations, particularly mapping class groups of fiber surfaces, Ozbagci and Stipsicz [26] constructed an infinite family of contact 3-manifolds each of which admits infinitely many pairwise homotopy inequivalent Stein fillings (see also [3,4,5,10,34]).…”

Surfaces in $D^4$ with the same boundary and fundamental group

“…This paper is concerned with symplectic surfaces in the standard symplectic 4-disk (D 4 , ω st ) bounded by the same transverse link in the standard contact 3-sphere (S 3 , ξ st ). Such surfaces have been studied in some papers [7,6,2,4].…”