The Johnson homomorphism and the third rational cohomology group of the Torelli group

Sakasai, Takuya

doi:10.1016/j.topol.2004.08.002

Cited by 19 publications

(29 citation statements)

References 10 publications

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“…1 A fixed symplectic basis for H 1 ( g,1 , F 2 ) Then it is straightforward to verify (see e.g. Lemma 2.1 of [22]) that…”

Section: Abelian Cyclesmentioning

confidence: 99%

“…However, the tools and techniques we are about to discuss will extend to the case of the full Torelli group. H 2 (K g,1 , F 2 ) In this section, we will give a method for constructing nontrivial classes in H 2 (K g,1 , F 2 ), inspired by Sakasai [22]. The idea is to construct abelian cycles in H 2 (K g,1 , F 2 ) and then to show that their images under the induced map…”

Section: Explicit Formulas On Generatorsmentioning

confidence: 99%

“…Johnson proved in [14] that the induced map in cohomology τ * : H * (∧ 3 H, Q) → H * (I g,1 , Q) (2) is an isomorphism in degree one. Hain [8] determined the image (and kernel) of τ * in degree two, as did Sakasai [22] (up to a small unknown) in degree three.…”

Section: Introductionmentioning

confidence: 99%

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The Birman–Craggs–Johnson homomorphism and abelian cycles in the Torelli group

Brendle

Farb

2007

Math. Ann.

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In the 1970s, Birman-Craggs-Johnson (BCJ) (Trans AMS 237: 283-309, 1978; Trans AMS 261(1):423-422, 1980) used Rochlin's invariant for homology 3-spheres to construct a remarkable surjective homomorphism σ : I g,1 → B 3 , where I g,1 is the Torelli group and B 3 is a certain F 2 -vector space of Boolean (square-free) polynomials. By pulling back cohomology classes and evaluating them on abelian cycles, we construct 2g 4 +O(g 3 ) dimensions worth of nontrivial elements of H 2 (I g,1 , F 2 ) which cannot be detected rationally. These classes in fact restrict to nontrivial classes in the cohomology of the subgroup K g,1 <

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“…1 A fixed symplectic basis for H 1 ( g,1 , F 2 ) Then it is straightforward to verify (see e.g. Lemma 2.1 of [22]) that…”

Section: Abelian Cyclesmentioning

confidence: 99%

Section: Explicit Formulas On Generatorsmentioning

confidence: 99%

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The Birman–Craggs–Johnson homomorphism and abelian cycles in the Torelli group

Brendle

Farb

2007

Math. Ann.

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“…The work of Igusa [38] (in particular Corollary 8.5.17) shows a close connection between the above problem with another very important problem (see Problem 11 in § 4) of non-triviality of Igusa's higher Franz-Reidemeister torsion classes in H 4i (IOut n ; R) (Igusa uses the notation Out h F n for the group IOut n ). We also refer to a recent work of Sakasai [95] which is related to the above problem.…”

Section: Higher Geometry Of the Mapping Class Groupmentioning

confidence: 99%

Cohomological structure of the mapping class group and beyond

Morita¹

2006

Proceedings of Symposia in Pure Mathematics

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Abstract. 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.

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“…Our previous paper [24] treated the third cohomology of the Torelli group. Brendle-Farb [6] studied the second cohomology of the Torelli group and the Johnson kernel by using the Birman-Craggs-Johnson homomorphism, and Pettet [23] studied the second cohomology of the (outer-)automorphism group of a free group by using its first Johnson homomorphism.…”

Section: Introductionmentioning

confidence: 99%

The second Johnson homomorphism and the second rational cohomology of the Johnson kernel

Sakasai

2007

Math. Proc. Camb. Phil. Soc.

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ABSTRACT. 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. In terms of the representation theory of the symplectic group, we give a complete description of cup products of two classes in the first rational cohomology of the Johnson kernel obtained by the rational dual of the second Johnson homomorphism.

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The Johnson homomorphism and the third rational cohomology group of the Torelli group

Cited by 19 publications

References 10 publications

The Birman–Craggs–Johnson homomorphism and abelian cycles in the Torelli group

The Birman–Craggs–Johnson homomorphism and abelian cycles in the Torelli group

Cohomological structure of the mapping class group and beyond

The second Johnson homomorphism and the second rational cohomology of the Johnson kernel

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