Abstract

We present three new inequalities tying the signature, the simplicial volume and the Euler characteristic of surface bundles over surfaces. Two of them are true for any surface bundle, while the third holds on a specific family of surface bundles, namely the ones that arise through ramified coverings. These are among the main known examples of bundles with non-zero signature.

Highlights

  • Surface bundles over surfaces form an interesting family of 4-manifolds that give rise to several questions: for example, do such manifolds with non-zero signature exist? If yes, which values does the signature take? What are the minimal base and fibre genera required to achieve a given signature? The relations and inequalities between signature and Euler characteristic of surface bundles have been widely studied, notably by Bryan, Catanese, Donagi, Endo, Korkmaz, Kotschick, Ozbagci, Rollenske, Stipsicz [5,6,10,11,12,17]

  • In the present note we add the comparison to the simplicial volume of the total space, using tools from bounded cohomology

  • The simplicial volume can act as a bridge between the two other invariants, signature and Euler characteristic

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Summary

Introduction

Surface bundles over surfaces form an interesting family of 4-manifolds that give rise to several questions: for example, do such manifolds with non-zero signature exist? If yes, which values does the signature take? What are the minimal base and fibre genera required to achieve a given signature? The relations and inequalities between signature and Euler characteristic of surface bundles have been widely studied, notably by Bryan, Catanese, Donagi, Endo, Korkmaz, Kotschick, Ozbagci, Rollenske, Stipsicz [5,6,10,11,12,17]. For any surface bundle E over a surface, the best known inequality between the signature σ (E) and the Euler characteristic χ(E) is due to Kotschick [17]: 2|σ (E)| ≤ χ(E). We compare here the signature to the simplicial volume of general surface bundles over surfaces and obtain: Theorem 1.1 Let E be an oriented surface bundle over a surface, with closed oriented base and fibre. We can give a lower bound on E under the form of the 1-norm of a distinguished 2-homology class: Proposition 1.2 Let E be an oriented surface bundle over a surface, with closed oriented base and fibre. Signatures remain, analogously to simplicial volume, quite hard to calculate for general surface bundles and are essentially only computed for bundles coming from specific constructions: differences of Lefschetz fibrations or ramified coverings.

Definition of the invariants π
Signature
Simplicial volume
The Euler class π
Construction of surface bundles using ramified coverings
Simplicial volume of such bundles
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