Jeremy Van Horn-Morris

Planar open books, monodromy factorizations and symplectic fillings

We study fillings of contact structures supported by planar open books by analyzing positive factorizations of their monodromy. Our method is based on Wendls theorem on symplectic fillings of planar open books. We prove that every virtually overtwisted contact structure on L(p,1) has a unique filling, and describe fillable and non-fillable tight contact structures on certain Seifert fibered spaces.

Monodromy substitutions and rational blowdowns

We introduce several new families of relations in the mapping class groups of planar surfaces, each equating two products of right-handed Dehn twists. The interest of these relations lies in their geometric interpretation in terms of rational blowdowns of 4-manifolds, specifically via monodromy substitution in Lefschetz fibrations. The simplest example is the lantern relation, already shown by the first author and Gurtas (‘Lantern relations and rational blowdowns’, Proc. Amer. Math. Soc. 138 (2010) 1131–1142) to correspond to rational blowdown along a−4 sphere; here we give relations that extend that result to realize the ‘generalized’ rational blowdowns of Fintushel and Stern (‘Rational blowdowns of smooth 4-manifolds’, J. Differential Geom. 46 (1997) 181–235) and Park (‘Seiberg–Witten invariants of generalised rational blow-downs’, Bull. Austral. Math. Soc. 56 (1997) 363–384) by monodromy substitution, as well as several of the families of rational blowdowns discovered by Stipsicz, Szabo, and Wahl (‘Rational blowdowns and smoothings of surface singularities’, J. Topol. 1 (2008) 477–517).

Fillings of unit cotangent bundles

We study the topology of exact and Stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface

Cabling, contact structures and mapping class monoids

\Sigma _g

Families of contact 3-manifolds with arbitrarily large Stein fillings

Σg, where g is at least 2. In particular, we prove a uniqueness theorem asserting that any Stein filling must be s-cobordant rel boundary to the disk cotangent bundle of

Monoids in the mapping class group

\Sigma _g