Right-handed Dehn Twists Research Articles

Mixed tête-à-tête twists as monodromies associated with holomorphic function germs

T\^ete-\`a-t\^ete graphs were introduced by N. A'Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed t\^ete-\`a-t\^ete graphs provide a generalization which define mixed t\^ete-\`a-t\^ete twists, which are pseudo-periodic automorphisms on surfaces. We characterize the mixed t\^ete-\`a-t\^ete twists as those pseudo-periodic automorphisms that have a power which is a product of right-handed Dehn twists around disjoint simple closed curves, including all boundary components. It follows that the class of t\^ete-\`a-t\^ete twists coincides with that of monodromies associated with reduced function germs on isolated complex surface singularities.

The diffeomorphism type of symplectic fillings

We show that simply connected contact manifolds that are subcritically Stein fillable have a unique symplectically aspherical filling up to diffeomorphism. Various extensions to manifolds with non-trivial fundamental group are discussed. The proof rests on homological restrictions on symplectic fillings derived from a degree-theoretic analysis of the evaluation map on a suitable moduli space of holomorphic spheres. Applications of this homological result include a proof that compositions of right-handed Dehn twists on Liouville domains are of infinite order in the symplectomorphism group. We also derive uniqueness results for subcritical Stein fillings up to homotopy equivalence and, under some topological assumptions on the contact manifold, up to diffeomorphism or symplectomorphism.

Symplectic mapping class group relations generalizing the chain relation

In this paper, we examine mapping class group relations of some symplectic manifolds. For each [Formula: see text] and [Formula: see text], we show that the [Formula: see text]-dimensional Weinstein domain [Formula: see text], determined by the degree [Formula: see text] homogeneous polynomial [Formula: see text], has a Boothby–Wang type boundary and a right-handed fibered Dehn twist along the boundary that is symplectically isotopic to a product of right-handed Dehn twists along Lagrangian spheres. We also present explicit descriptions of the symplectomorphisms in the case [Formula: see text] recovering the classical chain relation for the torus with two boundary components.

Stein fillable contact 3–manifolds and positive open books of genus one

A two-dimensional open book (S,h) determines a closed, oriented three-manifold Y(S,h) and a contact structure C(S,h) on Y(S,h). The contact structure C(S,h) is Stein fillable if h is positive, i.e. h can be written as a product of right-handed Dehn twists. Work of Wendl implies that when S has genus zero the converse statement holds, that is if C(S,h) is Stein fillable then h is positive. On the other hand, results by Wand and by Baker, Etnyre and Van Horn-Morris imply the existence of counterexamples to the converse statement with S of arbitrary genus strictly greater than one. The main purpose of this paper is to prove the converse statement under the assumption that S is a one-holed torus and Y(S,h) is a Heegaard Floer L-space.

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, Szabó, and Wahl (‘Rational blowdowns and smoothings of surface singularities’, J. Topol. 1 (2008) 477–517).

On sections of elliptic fibrations

We find a new relation among right-handed Dehn twists in the mapping class group of a $k$-holed torus for $4 \leq k \leq 9$. This relation induces an elliptic Lefschetz pencil structure on the four-manifold \cp $#(9-k)$ \cpb $ $ with $k$ base points and twelve singular fibers. By blowing up the base points we get an elliptic Lefschetz fibration on the complex elliptic surface $E(1)=$ \cp $#9$ \cpb $ \to S^2$ with twelve singular fibers and $k$ disjoint sections. More importantly we can locate these $k$ sections in a Kirby diagram of the induced elliptic Lefschetz fibration. The $n$-th power of our relation gives an explicit description for $k$ disjoint sections of the induced elliptic fibration on the complex elliptic surface $E(n) \to S^2$ for $n \geq 1$.

On a minimal factorization conjecture

Let ϕ : S → D be a proper holomorphic map from a connected complex surface S onto the open unit disk D ⊂ C , with 0 ∈ D as its unique singular value, and having fiber genus g > 0 . Assume that in case g ⩾ 2 , ϕ : S → D admits a deformation ϕ ′ : S ′ → D whose singular fibers are all of simple Lefschetz type. It has been conjectured that the factorization of the monodromy f ∈ M g around ϕ −1 ( 0 ) in terms of right-handed Dehn twists induced by the monodromy of ϕ ′ : S ′ → D has the least number of factors among all possible factorizations of f as a product of right-handed Dehn twists in the mapping class group (see [M. Ishizaka, One parameter families of Riemann surfaces and presentations of elements of mapping class group by Dehn twists, J. Math. Soc. Japan 58 (2) (2006) 585–594]). In this article, the validity of this conjecture is established for g = 1 .

Five-Dimensional Contact SU(2)- and SO(3)-Manifolds and Dehn Twists

In the first part of this paper the five-dimensional contact SO(3)-manifolds are classified up to equivariant coorientation preserving contactomorphisms. The construction of such manifolds with singular orbits requires the use of generalized Dehn twists. We show as an application that all simply connected 5-manifolds with singular orbits are realized by a Brieskorn manifold with exponents (k,2,2,2). The standard contact structure on such a manifold gives right-handed Dehn twists, and a second contact structure defined in the article gives left-handed twists. In an appendix we also describe the classification of five-dimensional contact SU(2)-manifolds.

On the signature of a Lefschetz fibration coming from an involution

In this article we show that the signature of a Lefschetz fibration coming from a special involution as a product of right-handed Dehn twists depends only on the number of genus on the involution axis. We investigate the geography of such Lefschetz fibrations and we identify it with a blow up of a ruled surface. We also get a geography of the Lefschetz fibration coming from a finite order element of mapping class group as a composition of two special involutions.

Open book decompositions for contact structures on Brieskorn manifolds

In this paper, we give an open book decomposition for the contact structures on some Brieskorn manifolds, in particular for the contact structures of Ustilovsky. The decomposition uses right-handed Dehn twists as conjectured by Giroux.

Heegaard Floer homology of certain mapping tori

We calculate the Heegaard Floer homologies$HF^+(M,s) for mapping tori M associated to certain surface diffeomorphisms, where s is any Spin^c structure on M whose first Chern class is non-torsion. Let gamma and delta be a pair of geometrically dual nonseparating curves on a genus g Riemann surface Sigma_g, and let sigma be a curve separating Sigma_g into components of genus 1 and g-1. Write t-gamma, t_delta, and t_sigma for the right-handed Dehn twists about each of these curves. The examples we consider are the mapping tori of the diffeomorphisms t_gamma^m circ t_delta^n for m,n in Z and that of t_sigma^{+-1}.

A NOTE ON THURSTON-WINKELNKEMPER'S CONSTRUCTION OF CONTACT FORMS ON 3-MANIFOLDS

A contact structure on a closed oriented 3-manifold 3 is a completely nonintegrable plane field on it. The existence of a contact structure on 3 was first proved by Martinet [14] and then Lutz [12], [13] for each homotopy class of plane fields. A co-orientable contact structure is given as the kernel of a 1-form α with α ∧ α > 0 or < 0, which is called a positive or negative contact form respectively. Thurston and Winkelnkemper [17] deduced the existence of a contact form in an elegant way from Alexander’s theorem [1] on open-book decompositions. Eliashberg [2] showed that the most of these contact structures are, however, too flexible for geometrical interest (see §2). On the other hand, he completely characterized more historic examples related closely to the complex analysis in several variables, namely, the strictly pseudo-convex boundary of compact Stein surfaces ([4]). The symplectic fillability is one of its significant generalizations. Recently Loi and Piergallini [11] translated Eliashberg’s characterization of compact Stein surfaces into the language of Lefschetz fibration by using Gompf’s method [8]. They also showed that 3 is realizable as the boundary of a Stein surface if and only if it admits an open-book decomposition whose monodromy map is a composition of right-handed Dehn-twists. In this note, we improve Thurston-Winkelnkemper’s construction so that we obtain a symplectically fillable contact structure in the case where the monodromy map of a given open-book decomposition is a composition of right-handed Dehn-twists along mutually disjoint curves (Theorem 2). An interesting by-product of our construction is Theorem 3 in §4 which gives a deformation of symplectically fillable contact structures into a foliation with a Reeb component. Note that a foliation admitting a Reeb component itself is not symplectically fillable. I would like to thank the referee for noticing the importance of Theorem 3.