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

Abstract

Abstract. 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 nonhyperelliptic fibrations on algebraic surfaces of general type. IntroductionThe study of Lefschetz fibrations has turned out to be interesting and important by virtue of the remarkable works of Donaldson [10] and Gompf [17] which demonstrate a close relationship between symplectic 4-manifolds and Lefschetz fibrations.The geography problem of Lefschetz fibrations is one of the most interesting topics to be investigated. Originally, the geography problem for complex surfaces was to find minimal surfaces of general type with a prescribed pair of Chern numbers. Using the pair of Euler characteristic and signature instead of the pair of Chern numbers, we can consider also the geography problem of Lefschetz fibrations. The Euler characteristic of a given Lefschetz fibration is easily computed, while the signature is not. If the Lefschetz fibration is over the 2-sphere, Ozbagci [33] In the present paper we would like to propose a useful method of computing signatures of Lefschetz fibrations and give some new examples of non-holomorphic Lefschetz fibrations. We first introduce the notion of signature for relations in mapping class groups. The signature of a Lefschetz fibration over the 2-sphere is equal to the sum of the signatures for basic relations appearing in its monodromy. Combining explicit computations of signatures for basic relations with a technique of substituting positive relations, which is a generalization of a method of Fuller and Smith [37], we construct new examples of non-holomorphic Lefschetz fibrations of genus 3, 4 and 5 which violate lower bounds of the slope of non-hyperelliptic fibrations in algebraic geometry. This paper is organized as follows. In Section 2 we recall a well-known theorem of Hopf and some facts on 2-cocycles of groups. 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. 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]. ...

Low-slope Lefschetz fibrations

“…Another invariant of f is the signature σ(X) of X. There are various techniques to compute the signature: see [14,15,6,5,11]. Endo and Nagami [5] showed that the signature of a Lefschetz fibration can be calculated by using the signatures of relations contained in its monodromy.…”

“…There are various techniques to compute the signature: see [14,15,6,5,11]. Endo and Nagami [5] showed that the signature of a Lefschetz fibration can be calculated by using the signatures of relations contained in its monodromy. We use this method to calculate the signatures of Lefschetz fibrations we construct.…”

“…Hain conjectured that for g ≥ 2 every relatively minimal genus-g Lefschetz fibration over S 2 satisfies the slope inequality (cf. [5,Conjecture 4.12] and [1,Question 5.10]). In [12] and [13], several examples of Lefschetz fibrations violating this conjecture are constructed.…”

Untitled

“…From the study of divisors in moduli space, Smith [2001] showed that a genus-3 Lefschetz fibration over S 2 which was produced by Fuller is nonholomorphic. Endo and Nagami [2005] constructed some examples of nonholomorphic Lefschetz fibrations which violate lower bounds of the slope for nonhyperelliptic fibrations of genus 3, 4 and 5 from the results of Konno [1991; and Chen [1993]. Hirose [2010] also gave some examples of g = 3, 4.…”

Non-holomorphic surface bundles and Lefschetz fibrations

“…Matsumoto's fibrations, and in turn the construction of Ozbagci and Stipsicz were generalized by Korkmaz in [14] for any g ≥ 2. (Also see [21] for other g = 2, [7] for g = 3, 4, 5 examples, and [8,9] for families whose total spaces are in a fixed homeomorphism class for any g ≥ 3.) Although these generalized Matsumoto fibrations and the standard fiber sum operation have complex interpretations, the twisted fiber sum operation does not, which allows one to perform it carefully so as to obtain fibrations on fourmanifolds which cannot support complex structures.…”