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Abstract. For any closed oriented surface Σg of genus g ≥ 3, we prove the existence of foliated Σg-bundles over surfaces such that the signatures of the total spaces are non-zero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form ω on the fiber. We relate the cohomology class represented by the transverse symplectic form to a crossed homomorphism ˜Flux: Symp Σg→H 1 (Σg; R) which is an extension of the flux homomorphism Flux: Symp 0 Σg→H 1 (Σg; R) from the identity component Symp 0 Σg to the whole group Symp Σg of symplectomorphisms of Σg with respect to the symplectic form ω. 1. Statement of the main results Let Σg be a closed oriented surface of genus g. It is a classical result that, for any g ≥ 3, there exist oriented Σg-bundles over closed oriented surfaces such that the signatures of the total spaces are non-zero, see Kodaira [18] and Atiyah [1]. In this paper, we prove the existence of such bundles which, in addition to having non-zero signature, are flat, or foliated. This means that there exist codimension two foliations complementary to the fibers, which is equivalent to the existence of lifts of the holonomy homomorphisms from the mapping class group to the diffeomorphism group of the fiber. We will further show that such lifts can be chosen to preserve a prescribed area form, or equivalently a

Abstract. Each finite dimensional irreducible rational representation V of the symplectic group Sp 2g(Q) determines a generically defined local system V over the moduli space Mg of genus g smooth projective curves. We study H 2 (Mg; V) and the mixed Hodge structure on it. Specifically, we prove that if g ≥ 6, then the natural map IH 2 ( ˜ Mg; V) → H 2 (Mg; V) is an isomorphism where ˜ Mg is the Satake compactification of Mg. Using the work of Saito we conclude that the mixed Hodge structure on H 2 (Mg; V) is pure of weight 2 + r if V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H 2 (Mg; V) for 3 ≤ g < 6. Results of this article can be applied in the study of relations in the Torelli group Tg. The moduli space Mg of smooth projective curves of genus g is a quasiprojective variety over C. Its points correspond to isomorphism classes of smooth projective complex curves of genus g. It has only finite quotient

ABSTRACT. Making use of the extended flux homomorphism defined in [13] on the group Symp Σg of symplectomorphisms of a closed oriented surface Σg of genus g ≥ 2, we introduce new characteristic classes of foliated surface bundles with symplectic, equivalently area-preserving, total holonomy. These characteristic classes are stable with respect to g and we show that they are highly non-trivial. We also prove that the second homology of the group HamΣg of Hamiltonian symplectomorphisms of Σg, equipped with the discrete topology, is very large for all g ≥ 2. 1.

Abstract. We define a category Fat b whose objects are isomorphism classes of bordered fat graphs and show that its geometric realization is a classifying space for the bordered mapping class groups. We then construct a CW structure on this geometric realization with one cell per isomorphism classes of bordered fat graphs. Its cellular cochain complex gives a bordered graph complex which computes the integral cohomology of the bordered mapping class groups. We use this chain complex to compute the cohomology of the bordered mapping class group of a torus with a single boundary and to describe certain homological operations induced Miller’s double-loop structure. 1.

Moduli spaces arise in classification problems in algebraic geometry (and other areas of geometry) when, as is typically the case, there are not enough discrete invariants to classify objects up to isomorphism. In the case of nonsingular complex projective curves (or compact Riemann surfaces) the genus g is a discrete invariant which classifies the curve regarded as a topological surface, but does not determine its complex structure when g> 0. For each g ≥ 0 there is a moduli space Mg whose points correspond bijectively to isomorphism classes of nonsingular complex projective curves of genus g, and whose geometric structure reflects the way such curves can vary in families depending on parameters. The topology of these moduli spaces Mg and their compactifications has been studied for several decades, and important progress has been made recently on some longstanding questions concerning their cohomology. In his fundamental paper [93] Mumford considered some tautological cohomological classes κj ∈ H 2j (Mg) for j = 1, 2,... which extend naturally to the Deligne-Mumford compactification Mg. Much work on the cohomology of Mg has concentrated on its tautological ring, which is the subalgebra of its rational cohomology ring (or of its Chow ring) generated by these tautological classes.

Dedicated to Jim Stasheff on the occasion of his sixtieth birthday. Abstract. We show that for g ≥ 2k + 3 the k-rational homotopy type of the moduli space Mg,n of algebraic curves of genus g with n punctures is independent of g, and the space Mg,n is k-formal. This implies the existence of a limiting rational homotopy type of Mg,n as g → ∞ and the formality of it.

For p a prime number, let X0(p)Q be the modular curve, over Q, parametrizing isogenies of degree p between elliptic curves, and let J0(p)Q be its jacobian variety. Let f: X0(p)Q → J0(p)Q