Mapping Class Group Relations Research Articles

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.

Erratum to “Lefschetz pencils and divisors in moduli space”

Proposition 5.1 of the paper [2], which states a constraint on the self-intersection of a section of a genus two Lefschetz fibration in terms of the number of singular fibres, is incorrect. Indeed, the paper itself essentially contains a counterexample. Namely, the genus two Lefschetz fibration X ! S obtained as the fibre sum of those associated to the mapping class group relations .a1b1a2b2a3/ 6 D 1; .a1b1a2b2/ 10 D 1;

Mapping class group relations, Stein fillings, and planar open book decompositions

The aim of this paper is to use mapping class group relations to approach the ‘geography’ problem for Stein fillings of a contact 3-manifold. In particular, we adapt a formula of Endo and Nagami so as to calculate the signature of such fillings as a sum of the signatures of basic relations in the monodromy of a related open book decomposition. We combine this with a theorem of Wendl to show that, for any Stein filling of a contact structure supported by a planar open book decomposition, the sum of the signature and Euler characteristic depends only on the contact manifold. This gives a simple obstruction to planarity, which we interpret in terms of the existence of certain configurations of curves in a factorization of the monodromy. We use these techniques to demonstrate examples of non-planar structures that cannot be shown to be non-planar by other existing methods.

Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations

We introduce the notion of signature for relations in mapping class groups and show that the signature of a Lefschetz fibration over the 2-sphere is the sum of the signatures for basic relations contained in its monodromy. Combining explicit calculations of the signature cocycle with a technique of substituting positive relations, we give some new examples of non-holomorphic Lefschetz fibrations of genus 3 , 4 3, 4 and 5 5 which violate slope bounds for non-hyperelliptic fibrations on algebraic surfaces of general type.