A local signature for fibered 4-manifolds with a finite group action. (English) Zbl 1297.57051

Let \(\Sigma_g\) be a closed surface of genus \(g\). Let \(G\) be a finite group, and \(p: \Sigma_g \to S^2\) a \(G\)-covering, that is, a finite regular covering whose deck transformation group is isomorphic to \(G\). Let \(E\) and \(B\) be compact oriented manifolds of dimension \(4\) and \(2\), and \(f : E \to B\) a smooth surjective map. A triple \((f, E, B)\) is a fibered \(4\)-manifold of the \(G\)-covering \(p\) if it satisfies (i) \(\partial E = f^{-1}(\partial B)\), (ii) \(f : E \to B\) has finitely many critical values \(\{ b_l \}_{l=1}^{n}\) (the inverse image \(f^{-1} (b_l)\) is called a singular fiber), and the restriction of \(p\) to the complement of singular fibers is a smooth oriented \(\Sigma_g\)-bundle, (iii) the structure group of this \(\Sigma_g\)-bundle is contained in the centralizer \(C(p)\) of the deck transformation group of \(p\) in \(\mathrm{Diff}_+ \Sigma_g\), (iv) the natural \(G\)-action on the complement of singular fibers extends to a smooth action on \(E\).
In this paper, it is shown that, for a \(G\)-covering \(p : \Sigma_g \to S^2\) with at least three branch points, there is a function \(\sigma_{loc}\) from the set of singular fiber germs of fibered 4-manifolds of the \(G\)-covering \(p\) to \(\mathbb{Q}\), such that for any fibered \(4\)-manifolds \((f, E, B)\) of \(G\)-covering, the signature of \(E\) is described as the sum \(\mathrm{Sign}(E) = \sum_{l=1}^{n} \sigma_{loc} (f_l)\), where \(f_l\) is a singular fiber germs of \(f : E \to B\). The function \(\sigma_{loc}\) is explained in Theorem 1.1 (the main theorem of this paper) and proved in Section 3 and 4. For the symmetric mapping class group \(\mathcal{M}_g(p) = \pi_0 C(p)\), the cobounding function of the pull back of the Meyer cocyle is considered to describe the local signature for fibered \(4\)-manifolds of the \(G\)-covering \(p\) when the condition (iv) is ignored and \(\mathcal{M}_g(p)\) satisfies some condition in Section 5. When \(G\) is abelian, the generating set of \(\mathcal{M}_g(p)\) is given in Section 6. When \(G = \mathbb{Z}_d\), for a special type of singular fiber germs, a formula for \(\sigma_{loc}\) is given in Section 7.


57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
Full Text: DOI arXiv Euclid


[1] T. Arakawa and T. Ashikaga, Local splitting families of hyperelliptic pencils, I, Tohoku Math. J. 53 (2001), no. 3, 369-394. · Zbl 1081.14508
[2] T. Ashikaga and H. Endo, Various aspects of degenerate families of Riemann surfaces, Sugaku Expositions 19 (2006), no. 2, 171-196. · Zbl 1359.32013
[3] T. Ashikaga and K. Konno, Global and local properties of pencils of algebraic curves, Adv. Stud. Pure Math. 36, Algebraic Geometry 2000 Azumino, 1-49, Math. Soc. Japan, Tokyo, 2002. · Zbl 1088.14010
[4] M. F. Atiyah, The logarithm of the Dedekind \(\eta\)-function, Math. Ann. 278 (1987), no. 1-4, 335-380. · Zbl 0648.58035
[5] M. F. Atiyah and I. M. Singer, The index of elliptic operators: III, Ann. of Math. (2) 87 (1968), 546-604. · Zbl 0164.24301
[6] J. S. Birman, Braids, links, and mapping class groups, Ann. of Math. Studies, No. 82, Princeton Univ. Press, Princeton, N.J., 1974.
[7] J. S. Birman and H. M. Hilden, On isotopies of homeomorphisms of Riemann surfaces, Ann. of Math. (2) 97 (1973), 424-439. · Zbl 0237.57001
[8] P. E. Conner, Differentiable periodic maps. Second edition, Lecture Notes in Math. 738, Springer, Berlin, 1979. · Zbl 0417.57019
[9] H. Endo, Meyer’s signature cocycle and hyperelliptic fibrations, Math. Ann. 316 (2000), no. 2, 237-257. · Zbl 0948.57013
[10] M. Furuta, Surface bundles and local signatures, Topological Studies around Riemann Surfaces (1999), 47-53. (Japanese)
[11] C. M. Gordon, On the \(G\)-signature theorem in dimension four, À la recherche de la topologie perdue, 159-180, Progr. Math. 62, Birkhäuser Boston, Boston, MA, 1986.
[12] F. Hirzebruch, The signature of ramified coverings, Global Analysis, 253-265, Univ. Tokyo Press and Princeton Univ. Press, Tokyo and Princeton, 1969. · Zbl 0208.51802
[13] F. Hirzebruch and D. Zagier, The Atiyah-Singer theorem and elementary number theory, Mathematics Lecture Series, No. 3, Publish or Perish, Inc., Boston, Mass., 1974. · Zbl 0288.10001
[14] S. Iida, Adiabatic limit of \(\eta\)-invariants and the Meyer function of genus two, Master’s thesis, University of Tokyo, 2004.
[15] N. Kawazumi and T. Uemura, Riemann-Hurwitz formula for Morita-Mumford classes and surface symmetries, Kodai Math. J. 21 (1998), 372-380. · Zbl 0924.57031
[16] Y. Kuno, The mapping class group and the Meyer function for plane curves, Math. Ann. 342 (2008), no. 4, 923-949. · Zbl 1162.57018
[17] Y. Kuno, The Meyer functions for projective varieties and their application to local signatures for fibered 4-manifolds, Algebr. Geom. Topol. 11 (2011), 145-195. · Zbl 1257.57027
[18] Y. Matsumoto, On 4-manifolds fibered by tori II, Proc. Japan Acad. 59 (1983), 100-103. · Zbl 0532.55020
[19] Y. Matsumoto, Lefschetz fibrations of genus two–a topological approach, Topology and Teichmüller spaces (Katinkulta, 1995), 123-148, World Sci. Publ., River Edge, NJ, 1996. · Zbl 0921.57006
[20] W. Meyer, Die Signatur von Flächenbündeln, Math. Ann. 201 (1973), no. 3, 239-264. · Zbl 0241.55019
[21] T. Morifuji, On Meyer’s function of hyperelliptic mapping class groups, J. Math. Soc. Japan 55 (2003), no. 1, 117-129. · Zbl 1031.57017
[22] S. Morita, Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles I, Topology 28 (1989), no. 3, 305-323. · Zbl 0684.57008
[23] T. Nakata, On a local signature of Lefschetz singular fibers in hyperelliptic fibrations, Master’s thesis, University of Tokyo, 2005. (Japanese)
[24] M. Namba and M. Takai, Degenerating families of finite branched coverings, Osaka J. Math. 40 (2003), no. 1, 139-170. · Zbl 1026.32025
[25] M. Sato, The abelianization of a symmetric mapping class group, Math. Proc. Cambridge Philos. Soc. 147 (2009), no. 2, 369-388. · Zbl 1177.57017
[26] V. G. Turaev, First symplectic Chern class and Maslov indices, J. Math. Sci. 37 (1987), no. 3, 1115-1127. · Zbl 0612.55019
[27] K. Ueno, Discriminants of curves of genus 2 and arithmetic surfaces, Algebraic geometry and commutative algebra, Vol. II, 749-770, Kinokuniya, Tokyo, 1988. · Zbl 0707.14025
[28] K. Yoshikawa, A local signature for generic 1-parameter deformation germs of a complex curve, Algebraic Geometry and Topology of Degenerations, Coverings and Singularities (2000), 188-200. (Japanese)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.