Extended flux maps on surfaces and the contracted Johnson homomorphism
نویسندگان
چکیده
منابع مشابه
Extended flux maps on surfaces and the contracted Johnson homomorphism
On a closed symplectic surface Σ of genus two or more, we give a new construction of an extended flux map (a crossed homomorphism from the symplectomorphism group Symp(Σ) to the cohomology group H(Σ;R) that extends the flux homomorphism). This construction uses the topology of the Jacobian of the surface and a correction factor related to the Johnson homomorphism. For surfaces of genus three or...
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We give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general result that also applies to “subsurface Torelli groups”. Using this, we extend Johnson’s calculation of the rational abelianization of the Torelli group not on...
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Let Σg → E → Σh be a surface bundle over a surface with monodromy representation ρ : π1Σh → Mod(Σg) contained in the Torelli group Ig. In this paper we express the cup product structure in H∗(E,Z) in terms of the Johnson homomorphism τ : Ig → ∧(H1(Σg ,Z)). This is applied to the question of obtaining an upper bound on the maximal n such that p1 : E → Σh1 , . . . , pn : E → Σhn are fibering maps...
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Let Fn be the free group on n generators. Define IAn to be group of automorphisms of Fn that act trivially on first homology. The Johnson homomorphism in this setting is a map from IAn to its abelianization. The first goal of this paper is to determine how much this map contributes to the second rational cohomology of IAn . A descending central series of IAn is given by the subgroups K (i) n wh...
متن کاملThe Second Johnson Homomorphism and the Second Rational Cohomology of the Johnson Kernel
The Johnson kernel is the subgroup of the mapping class group of a surface generated by Dehn twists along bounding simple closed curves, and has the second Johnson homomorphism as a free abelian quotient. We will determine the kernel of the map induced on the second rational cohomology by the second Johnson homomorphism in terms of the representation theory of the symplectic group.
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ژورنال
عنوان ژورنال: Journal of Symplectic Geometry
سال: 2011
ISSN: 1527-5256,1540-2347
DOI: 10.4310/jsg.2011.v9.n4.a3