On the structure of the torelli group and the casson invariant

Morita, Shigeyuki

doi:10.1016/0040-9383(91)90042-3

Cited by 66 publications

(96 citation statements)

References 11 publications

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“…Specifically, these invariants are expressed in (6) of Theorem 3, in (7) of Theorem 4, in (15) of Theorem 6, and in (18) of Theorem 7 as traces and matrix elements of operators acting on * H 1 (Σ) for a surface Σ. We relate these formulae to previous results in [10], [11], [12] and [5] on the Frohman-Nicas and Reshetikhin-Turaev theories.…”

Section: Introductionmentioning

confidence: 87%

p–Modular TQFT's, Milnor torsion and the Casson–Lescop invariant

Kerler¹

2002

Invariants of Knots and 3--Manifolds (Kyoto 2001)

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We derive formulae which lend themselves to TQFT interpretations of the Milnor torsion, the Lescop invariant, the Casson invariant, and the Casson-Morita cocyle of a 3-manifold, and, furthermore, relate them to the Reshetikhin-Turaev theory. AMS Classification Introduction and summaryInvariants of 3-manifolds that admit extensions to topological quantum field theories (TQFT) are structurally highly organized. Consequently, their evaluations permit an equally deeper insight into the topological structure of the underlying 3-manifolds beyond the mere distinction of their homeomorphism types. Although the notion and many examples of TQFT's have been around for more than a decade there are still surprisingly large gaps in the understanding of the explicit TQFT content of the "classical" invariants such as the Milnor-Turaev torsion or the Casson-Walker-Lescop invariant.In this article we derive formulae which lend themselves to TQFT interpretations of the Milnor torsion, the Lescop invariant, the Casson invariant, and the Casson-Morita cocyle of a 3-manifold. Specifically, these invariants are expressed in (6) of Theorem 3, in (7) of Theorem 4, in (15) of Theorem 6, and in (18) of Theorem 7 as traces and matrix elements of operators acting on * H 1 (Σ) for a surface Σ. We relate these formulae to previous results in [10],[11], [12] and [5] on the Frohman-Nicas and Reshetikhin-Turaev theories. In the course, we develop the general notion of a q/l-solvable TQFT and consider reductions to the p-modular cases, as needed for the quantum theories. As an

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Section: Introductionmentioning

confidence: 87%

p–Modular TQFT's, Milnor torsion and the Casson–Lescop invariant

Kerler¹

2002

Invariants of Knots and 3--Manifolds (Kyoto 2001)

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“…Mg the class d i is defined only for g ≥ 12i − 9 at present, although it is highly likely that it is defined for all g. It was proved in [75] …”

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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“…The bracket operation of h g,1 is explicitly given in [19,20]. In this paper, however, we use an alternative description given by Garoufalidis-Levine [8], Levine [14,15], which will be easier to handle.…”

Section: Remark 22mentioning

confidence: 99%

“…In each case, our task is divided into the following two parts. First, we will find some summands in Lemma 4.1 (or that corresponding to each case) which belong to the kernel by using Stallings' exact sequence in [25] together with Morita's description [19,20] of Johnson's homomorphisms as a Lie algebra homomorphism. Then we show that the other summands actually survive in the second cohomology by constructing explicit cycles which come from abelian subgroups of the Johnson kernel and give non-trivial values by the Kronecker product.…”

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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On the structure of the torelli group and the casson invariant

Cited by 66 publications

References 11 publications

p–Modular TQFT's, Milnor torsion and the Casson–Lescop invariant

p–Modular TQFT's, Milnor torsion and the Casson–Lescop invariant

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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