The extension of Johnson's homomorphism from the Torelli group to the mapping class group

Morita, Shigeyuki

doi:10.1007/bf01231286

Cited by 96 publications

(104 citation statements)

References 17 publications

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“…The above 1-cocycle j is M g, * -equivariant and we can show that τ 1 is the restriction of a crossed homomorphism from M g, * to Λ 3 H. This gives a new proof that τ 1 so extends, as first shown in [29].…”

Section: Introductionmentioning

confidence: 75%

“…constructed in [29]. By Corollary 3.4 and Proposition 4.3, the proof would be complete if M g, * were an Eilenberg-MacLane space of type K(M g, * , 1).…”

Section: Proof Of Theorem 42mentioning

confidence: 89%

“…It was shown in [29] that the Johnson homomorphism τ 1 : I g, * −→Λ 3 H can be extended to a crossed homomorphismk : M g, * −→ 1 2 Λ 3 H essentially uniquely. Here we prove that it in fact lifts to a canonical 1-cocycle j of the cell complexĜ T with values in Λ 3 H which is M g, * -equivariant.…”

Section: Combinatorial Realization Of the Extended Johnson Homomorphismmentioning

confidence: 99%

“…In [29], there was constructed a crossed homomorphism k : M g, * −→ 1 2 Λ 3 H whose restriction to the Torelli group coincides with the Johnson homomorphism τ 1 : I g, * →Λ 3 H.…”

Section: Combinatorial Realization Of the Extended Johnson Homomorphismmentioning

confidence: 99%

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Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves

Morita¹,

Penner²

2008

Math. Proc. Camb. Phil. Soc.

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Abstract. Infinite presentations are given for all of the higher Torelli groups of once-punctured surfaces. In the case of the classical Torelli group, a finite presentation of the corresponding groupoid is also given, and finite presentations of the classical Torelli groups acting trivially on homology modulo N are derived for all N . Furthermore, the first Johnson homomorphism, which is defined from the classical Torelli group to the third exterior power of the homology of the surface, is shown to lift to an explicit canonical 1-cocycle of the Teichmüller space. The main tool for these results is the known mapping class group invariant ideal cell decomposition of the Teichmüller space.This new 1-cocycle is mapping class group equivariant, so various contractions of its powers yield various combinatorial (co)cycles of the moduli space of curves, which are also new. Our combinatorial construction can be related to former works of Kawazumi and the first-named author with the consequence that the algebra generated by the cohomology classes represented by the new cocycles is precisely the tautological algebra of the moduli space.There is finally a discussion of prospects for similarly finding cocycle lifts of the higher Johnson homomorphisms.

show abstract

Section: Introductionmentioning

confidence: 75%

“…constructed in [29]. By Corollary 3.4 and Proposition 4.3, the proof would be complete if M g, * were an Eilenberg-MacLane space of type K(M g, * , 1).…”

Section: Proof Of Theorem 42mentioning

confidence: 89%

Section: Combinatorial Realization Of the Extended Johnson Homomorphismmentioning

confidence: 99%

“…In [29], there was constructed a crossed homomorphism k : M g, * −→ 1 2 Λ 3 H whose restriction to the Torelli group coincides with the Johnson homomorphism τ 1 : I g, * →Λ 3 H.…”

Section: Combinatorial Realization Of the Extended Johnson Homomorphismmentioning

confidence: 99%

See 2 more Smart Citations

Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves

Morita¹,

Penner²

2008

Math. Proc. Camb. Phil. Soc.

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“…[20]). In our papers [77][81], we constructed a morphism [81] for details). The diagram (6) induces the following commutative diagram.…”

Section: Tautological Algebra 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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Cup products in surface bundles, higher Johnson invariants, and MMM classes

Salter

2017

Math. Z.

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Abstract. In this paper we prove a family of results connecting the problem of computing cup products in surface bundles to various other objects that appear in the theory of the cohomology of the mapping class group Modg and the Torelli group Ig. We show that N. Kawazumi's twisted MMM class m 0,k can be used to compute k-fold cup products in surface bundles, and that m 0,k provides an extension of the higher Johnson invariant τ k−2to H k−2 (Modg, * , ∧ k H 1 ). These results are used to show that the behavior of the restriction of the even MMM classes e 2i to H 4i (I 1 g ) is completely determined by Im(τ 4i ) ≤ ∧ 4i+2 H 1 , and to give a partial answer to a question of D. Johnson. We also use these ideas to show that all surface bundles with monodromy in the Johnson kernel Kg, * have cohomology rings isomorphic to that of a trivial bundle, implying the vanishing of all τ i when restricted to Kg, * .

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The extension of Johnson's homomorphism from the Torelli group to the mapping class group

Cited by 96 publications

References 17 publications

Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves

Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves

Cohomological structure of the mapping class group and beyond

Cup products in surface bundles, higher Johnson invariants, and MMM classes

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