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25
Mapping class groups and moduli spaces of curves
 Proc. Symposia in Pure Math
, 1997
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The primary approximation to the cohomology of the moduli space of curves and cocycles for the MumfordMoritaMiller classes
, 2001
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COHOMOLOGICAL STRUCTURE OF THE MAPPING CLASS GROUP AND BEYOND
, 2005
"... In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer au ..."
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Cited by 22 (3 self)
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In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer automorphism group of free groups, the symplectomorphism group of surfaces. Based on these, we present several conjectures and problems concerning the cohomology of these groups. We are particularly interested in the possible interplays between these cohomology groups rather than merely the structures of individual groups. It turns out that, we have to include, in our considerations, two other groups which contain the mapping class group as their core subgroups and whose structures seem to be deeply related to that of the mapping class group. They are the arithmetic mapping class group and the group of homology cobordism classes of homology cylinders. 1
Signatures of foliated surface bundles and the symplectomorphism groups of surfaces
, 2003
"... For any closed oriented surface Σg of genus g ≥ 3, we prove the existence of foliated Σgbundles over surfaces such that the signatures of the total spaces are nonzero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form ω on the fiber. We relat ..."
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Cited by 17 (4 self)
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For any closed oriented surface Σg of genus g ≥ 3, we prove the existence of foliated Σgbundles over surfaces such that the signatures of the total spaces are nonzero. 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.
The second cohomology with symplectic coefficients of the moduli space of smooth projective curves
 Compositio Math
, 1998
"... 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 ..."
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Cited by 6 (0 self)
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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
Characteristic classes of foliated surface bundles with areapreserving total holonomy, preprint
"... 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 areapreserving, total holonomy. These c ..."
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Cited by 6 (3 self)
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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 areapreserving, total holonomy. These characteristic classes are stable with respect to g and we show that they are highly nontrivial. 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.
Stable Cohomology of the Universal Picard Varieties and the Extended Mapping Class Group
 DOCUMENTA MATH.
, 2012
"... We study the moduli spaces which classify smooth surfaces along with a complex line bundle. There are homological stability and Madsen–Weiss type results for these spaces (mostly due to Cohen and Madsen), and we discuss the cohomological calculations which may be deduced from them. We then relate ..."
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Cited by 6 (0 self)
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We study the moduli spaces which classify smooth surfaces along with a complex line bundle. There are homological stability and Madsen–Weiss type results for these spaces (mostly due to Cohen and Madsen), and we discuss the cohomological calculations which may be deduced from them. We then relate these spaces to (a generalisation of) Kawazumi’s extended mapping class groups, and hence deduce cohomological information about these. Finally, we relate these results to complex algebraic geometry. We construct a holomorphic stack classifying families of Riemann surfaces equipped with a fibrewise holomorphic line bundle, which is a gerbe over the universal Picard variety, and compute its holomorphic Picard group.