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 non-zero 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.Comment: 20 pages, 7 figures, accepted for publication in Topolog

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

“…As a consequence, we conclude that signatures of Lefschetz fibrations corresponding to relators obtained in §3 of [13] do not depend on topological types of relators. 2 of genus g is a positive relator, we obtain…”

“…The idea of "the signature of a relation" occurred to the first author when he worked on signature of surface bundles at the University of Munich in 2000 with the co-authors of [13]. He is grateful to the co-authors of [13], especially D. Kotschick for helpful discussions and comments.…”

“…He is grateful to the co-authors of [13], especially D. Kotschick for helpful discussions and comments. The authors are grateful to K. Konno for a detailed explanation of his works on the lower bound of the slope of non-hyperelliptic fibrations and to T. Morifuji for useful comments on his formula of Meyer's function in [30].…”

“…In Section 3 we define the signature of a relator of mapping class groups and carry out explicit computations of signatures for basic relators. In Section 4 we construct examples of non-holomorphic Lefschetz fibrations using the results of Section 3 and some results on slope bounds for algebraic surfaces.The idea of "the signature of a relation" occurred to the first author when he worked on signature of surface bundles at the University of Munich in 2000 with the co-authors of [13]. He is grateful to the co-authors of [13], especially D. Kotschick for helpful discussions and comments.…”

Minimal number of singular fibers in Lefschetz fibrations over the torus

“…Then each vanishing cycle except y 3 is nonseparating and Σ 3 − {δ 0 , x 2 } and Σ 3 − {δ 2 , x 1 } are connected, δ 0 is disjoint from x 2 , and δ 2 is disjoint from x 1 . As it is explained in [6,14], there is a map ψ : Σ 3 → Σ 3 satisfying ψ(δ 0 ) = x 1 , ψ(x 2 ) = δ 2 , implying that We get the same result for g ≥ 4, since we can cap off the two boundary components δ 0 and δ 2 by a twice punctured genus-(g − 3) surface.…”