Surface bundles over surfaces: new inequalities between signature, simplicial volume and Euler characteristic
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Surface bundles over surfaces: new inequalities between signature, simplicial volume and Euler characteristic Michelle Bucher1 · Caterina Campagnolo2 Received: 12 October 2018 / Accepted: 1 October 2020 © The Author(s) 2020
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. Keywords Surface bundle · Simplicial volume · Signature Mathematics Subject Classification Primary 55R10 · Secondary 57R22
1 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 [5,6,10–12,17]. 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 (for the definition of simplicial volume, see Sect. 2.2). 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).
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Caterina Campagnolo [email protected] Michelle Bucher [email protected]
1
Section de Mathématiques, Université de Genève, Geneva, Switzerland
2
Unité de Mathématiques Pures et Appliquées, ENS Lyon, Lyon, France
123
Geometriae Dedicata
Kotschick also obtained the stronger inequality 3|σ (E)| < χ(E) in some special cases [18]. The first author’s work on simplicial volume of surface bundles [8] produced an inequality between simplicial volume and Euler characteristic of aspherical surface bundles: 6χ(E) ≤ 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. Then 36|σ (E)| ≤ E. Observe that this is stronger than the combination of Kotschick’s and the first author’s inequalities, which only give 12|σ (E)| ≤ E, or 18|σ (E)| < E in the special cases of [18]. The inequality of Theorem 1.1 is also strictly stronger than the value produced by the up to now best example [10, Theorem A], which is 27|σ (E)| ≤ E. The simplicial volume remains very hard to compute explicitly. In fact, the exact values in non-vanishing cases are known only for hyperbolic manifolds (due to Gromov–Thurston [14,22]) and for locally (H2
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