Flat structures on surface bundles

Bowden, Jonathan¹

¹Institut für Mathematik, Universität Augsburg, Universitätsstr 14, D-86159 Augsburg, Germany

Algebraic and Geometric Topology, Tome 11 (2011) no. 4, pp. 2207-2235

Résumé

We show that there exist flat surface bundles with closed leaves having nontrivial normal bundles. This leads us to compute the abelianisation of surface diffeomorphism groups with marked points. We also extend a formula of Tsuboi that expresses the Euler class of a flat circle bundle in terms of the Calabi invariant of certain Hamiltonian diffeomorphisms to surfaces of higher genus and derive a similar formula for the first MMM–class of surface bundles with punctured fibre.

Détail

DOI : 10.2140/agt.2011.11.2207

Classification : 37E30, 57R30, 57R50, 57R17
Keywords: diffeomorphism group, mapping class group, foliation, symplectic topology, group cohomology, characteristic class of surface bundle

Bibliographie

[1] G Bande, D Kotschick, The geometry of symplectic pairs, Trans. Amer. Math. Soc. 358 (2006) 1643

[2] M Bestvina, T Church, J Souto, Some groups of mapping classes not realized by diffeomorphisms

[3] R Bott, Lectures on characteristic classes and foliations, from: "Lectures on algebraic and differential topology (Second Latin American School in Math., Mexico City, 1971)" (editor S Gitler), Lecture Notes in Math. 279, Springer (1972) 1

[4] J Bowden, Two-dimensional foliations on four-manifolds, PhD thesis, Ludwig-Maximillians-Universität München (2010)

[5] K Fukui, Homologies of the group $\mathrm{Diff}^{\infty }(\mathbf{R}^{n}, 0)$ and its subgroups, J. Math. Kyoto Univ. 20 (1980) 475

[6] N V Ivanov, Mapping class groups, from: "Handbook of geometric topology" (editors R J Daverman, R B Sher), North-Holland (2002) 523

[7] D Kotschick, S Morita, Signatures of foliated surface bundles and the symplectomorphism groups of surfaces, Topology 44 (2005) 131

[8] D Kotschick, S Morita, Characteristic classes of foliated surface bundles with area-preserving holonomy, J. Differential Geom. 75 (2007) 273

[9] D Mcduff, D Salamon, Introduction to symplectic topology, Oxford Math. Monogr., The Clarendon Press, Oxford Univ. Press (1998)

[10] J Milnor, On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1958) 215

[11] Y Mitsumatsu, On the self-intersections of foliation cycles, Trans. Amer. Math. Soc. 334 (1992) 851

[12] S Morita, Characteristic classes of surface bundles, Invent. Math. 90 (1987) 551

[13] S Morita, Families of Jacobian manifolds and characteristic classes of surface bundles. I, Ann. Inst. Fourier (Grenoble) 39 (1989) 777

[14] S Morita, Families of Jacobian manifolds and characteristic classes of surface bundles. II, Math. Proc. Cambridge Philos. Soc. 105 (1989) 79

[15] J Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (1978) 347

[16] S Sternberg, On the structure of local homeomorphisms of euclidean $n$–space. II, Amer. J. Math. 80 (1958) 623

[17] S Sternberg, The structure of local homeomorphisms. III, Amer. J. Math. 81 (1959) 578

[18] W Thurston, Foliations and groups of diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974) 304

[19] T Tsuboi, The Calabi invariant and the Euler class, Trans. Amer. Math. Soc. 352 (2000) 515