Hisaaki Endo

Bounded cohomology and non-uniform perfection of mapping class groups

Abstract.Using the existence of certain symplectic submanifolds in symplectic 4-manifolds, we prove an estimate from above for the number of singular fibers with separating vanishing cycles in minimal Lefschetz fibrations over surfaces of positive genus. This estimate is then used to deduce that mapping class groups are not uniformly perfect, and that the map from their second bounded cohomology to ordinary cohomology is not injective.

Linear independence of topologically slice knots in the smooth cobordism group

Abstract By using a result of M. Furuta concerning the homology cobordism group of homology 3-spheres, we give an infinite family of topologically slice knots which are linearly independent in the smooth knot-cobordism group.

Lantern relations and rational blowdowns

We discuss a connection between the lantern relation in mapping class groups and the rational blowing down process for 4-manifolds. More precisely, if we change a positive relator in Dehn twist generators of the mapping class group by using a lantern relation, the corresponding Lefschetz fibration changes into its rational blowdown along a copy of the configuration C 2 . We exhibit examples of such rational blowdowns of Lefschetz fibrations whose blowup is homeomorphic but not diffeomorphic to the original fibration.

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 and 5 which violate slope bounds for non-hyperelliptic fibrations on algebraic surfaces of general type.

Commutators, Lefschetz fibrations and the signatures of surface bundles ☆

Abstract We construct examples of Lefschetz fibrations with prescribed singular fibers. By taking differences of pairs of such fibrations with the same singular fibers, we obtain new examples of surface bundles over surfaces with nonzero signature. From these we derive new upper bounds for the minimal genus of a surface representing a given element in the second homology of a mapping class group.

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).

Failure of separation by quasi-homomorphisms in mapping class groups

We show that mapping class groups of surfaces of genus at least two contain elements of infinite order that are not conjugate to their inverses, but whose powers have bounded torsion lengths. In particular every homogeneous quasi-homomorphism vanishes on such an element, showing that elements of infinite order not conjugate to their inverses cannot be separated by quasi-homomorphisms.

A gap theorem for positive Einstein metrics on the four-sphere

We show that there exists a universal positive constant

Meyer's signature cocycle and hyperelliptic fibrations

\varepsilon _0 > 0