Casson's invariant for homology 3-spheres and characteristic classes of surface bundles I

Morita, Shigeyuki

doi:10.1016/0040-9383(89)90011-6

Cited by 97 publications

(139 citation statements)

References 18 publications

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“…Johnson [12] showed that K g coincides with the kernel of what is now called the first Johnson homomorphism τ g (1) of I g (see [10]), namely we have an exact sequence The group K g plays an important role in topology. For example, it has some relationships to the Casson invariant of homology 3-spheres and secondary characteristic classes of surface bundles as we see in Morita's papers [18,21]. However, we still do not have enough information on K g .…”

Section: Introductionmentioning

confidence: 97%

“…By results of Johnson [10] for k = 1, Hain [9] and Morita [18] for k = 2, Hain [9] and Asada-Nakamura [3] for k = 3, we have the following.…”

Section: Johnson's Homomorphisms Via the Representation Theory Ofmentioning

confidence: 99%

“…In [18], Morita computed the value of the Dehn twist ψ h ∈ K g,1 along the simple closed curve γ h bounding the surface Σ h,1 of the standard position by τ Q g,1 (2), and he obtained that…”

Section: Description Of the Second Johnson Homomorphismmentioning

confidence: 99%

“…In terms of the symplectic representation theory, h g (2) ⊗ Q is the irreducible representation denoted by [2 2 ] (see [9], [18]). By passing to the dual, we obtain an injection…”

Section: Introductionmentioning

confidence: 99%

“…Brendle-Farb [6] studied H 2 (K g,1 ; Z) by using the Birman-Craggs-Johnson homomorphism and its integral lift by Morita [18], and showed that the rank of H 2 (K g,1 ; Z) is at least 16g 4 +O(g 3 ). From our computation, we can give a sharper estimate.…”

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confidence: 99%

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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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Section: Introductionmentioning

confidence: 97%

“…By results of Johnson [10] for k = 1, Hain [9] and Morita [18] for k = 2, Hain [9] and Asada-Nakamura [3] for k = 3, we have the following.…”

Section: Johnson's Homomorphisms Via the Representation Theory Ofmentioning

confidence: 99%

Section: Description Of the Second Johnson Homomorphismmentioning

confidence: 99%

“…In terms of the symplectic representation theory, h g (2) ⊗ Q is the irreducible representation denoted by [2 2 ] (see [9], [18]). By passing to the dual, we obtain an injection…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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

Sakasai

2007

Math. Proc. Camb. Phil. Soc.

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

Morita

1993

Invent Math

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Morphology of the petal inAconitum

Kosuge

Tamura

1988

Bot Mag Tokyo

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The Chillingworth subgroup of the mapping class group of a compact oriented surface of genus g with one boundary component is defined as the subgroup whose elements preserve nonvanishing vector fields on the surface up to homotopy. In this work, we determine the rational abelianization of the Chillingworth subgroup as a full mapping class group module. The abelianization is given by the first Johnson homomorphism and the Casson-Morita homomorphism for the Chillingworth subgroup. Additionally, we compute the order of the Euler class of a certain central extension related to the Chillingworth subgroup and determine the kernel of the Casson-Morita homomorphism for the Chillingworth subgroup. Contents 1. Introduction 1 2. Preliminaries 3 2.1. The action of the mapping class group on the fundamental group of the surface and the Johnson homomorphisms 4 2.2. The space of tree diagrams and the infinitesimal Dehn-Nielsen representation 5 3. The action on the set of homotopy classes of vector fields on the surface and the Chillingworth subgroup 7 3.1. The first Johnson homomorphism and the Chillingworth subgroup 9 4. Determination of Im((τ g,1 (1)) * : H 2 (Ch g,1 ; Q) → H 2 (U ; Q)) and Ker((τ g,1 (1)) * : H 2 (U ; Q) → H 2 (Ch g,1 ; Q)) 10 5. The Casson-Morita homomorphism d : K g,1 → Z and its extension d : Ch g,1 → Z over the Chillingworth subgroup 17 6. Determination of H 1 (Ch g,1 ; Q), H 1 (Ch g, * ; Q), H 1 (Ch g ; Q) 19 References 23

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Casson's invariant for homology 3-spheres and characteristic classes of surface bundles I

Cited by 97 publications

References 18 publications

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

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

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

Morphology of the petal inAconitum

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