Abelian quotients of subgroups of the mapping class group of surfaces

Morita, Shigeyuki

doi:10.1215/s0012-7094-93-07017-2

Cited by 170 publications

(205 citation statements)

References 14 publications

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“…Natural first attempts would be induction and Gosper's algorithm, and neither of these is successful, as for one thing, the summands depend not only on the summing index i but also on n. The challenge remains to establish a closed-form expression for N q (m, n; ∞; ∞) and N q (m, n; ∞; q − 1) for higher m. Today, his homomorphism τ is called the first Johnson homomorphism and has been generalized to those of higher degrees. Over the last two decades, good progress was made in the study of the Johnson homomorphisms of mapping class groups through the work of many authors including Morita [1993a], Hain [1997] and others. Let F n be a free group generated by x 1 , x 2 , .…”

Section: Main Theoremsmentioning

confidence: 99%

Untitled

2015

Pacific J. Math.

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In this paper we give a combinatorial characterization of tight fusion frame (TFF) sequences using Littlewood-Richardson skew tableaux. The equal rank case has been solved recently by Casazza, Fickus, Mixon, Wang, and Zhou. Our characterization does not have this limitation. We also develop some methods for generating TFF sequences. The basic technique is a majorization principle for TFF sequences combined with spatial and Naimark dualities. We use these methods and our characterization to give necessary and sufficient conditions which are satisfied by the first three highest ranks. We also give a combinatorial interpretation of spatial and Naimark dualities in terms of Littlewood-Richardson coefficients. We exhibit four classes of TFF sequences which have unique maximal elements with respect to majorization partial order. Finally, we give several examples illustrating our techniques including an example of tight fusion frame which can not be constructed by the existing spectral tetris techniques. We end the paper by giving a complete list of maximal TFF sequences in dimensions ≤ 9.

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Section: Main Theoremsmentioning

confidence: 99%

Untitled

2015

Pacific J. Math.

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“…Let us now be more precise. We recall the definition of the Johnson filtration and the Johnson homomorphisms, which have been introduced and studied by Johnson and Morita in [10,22]. Recall that π := π 1 (Σ) is a free group.…”

Section: Introductionmentioning

confidence: 99%

Triviality of the $J_4$-equivalence among homology 3-spheres

Faes¹

2021

Preprint

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We prove that all homology 3-spheres are J 4 -equivalent, i.e. that any homology 3-sphere can be obtained from one another by twisting one of its Heegaard splittings by an element of the mapping class group acting trivially on the fourth nilpotent quotient of the fundamental group of the gluing surface. We do so by exhibiting an element of J 4 , the fourth term of the Johnson filtration of the mapping class group, on which (the core of) the Casson invariant takes the value 1. In particular, this provides an explicit example of an element of J 4 that is not a commutator of length 2 in the Torelli group. Contents 1. Introduction 1 2. A special element of J 4 4 3. Triviality of the J 4 -equivalence 14 Appendix A. SageMath computer program 15This research has been supported by the project "AlMaRe" (ANR-19-CE40-0001-01) of the ANR and the project "ITIQ-3D" of the Région Bourgogne Franche-Comté.The IMB receives support from the EIPHI Graduate School (contract ANR-17-EURE-0002).

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“…As stepwise approximations of ρ, we can consider the action of M on the nilpotent quotients of π ρ m : M −→ Aut(π/Γ m+1 π), where Γ 1 π := π and Γ m+1 π := [π, Γ m π] for m ≥ 1, define the lower central series of π. This is the approach pursued by D. Johnson [20] and S. Morita [36]. This approach allows to define the Johnson filtration…”

Section: Introductionmentioning

confidence: 99%

“…A symplectic derivation d of Lie(H) is a derivation satisfying d(Ω) = 0. S. Morita shows in [36] that for h ∈ J m M, the morphism τ m (h) defines a symplectic derivation of Lie(H). The group of symplectic degree m derivations of Lie(H) can be canonically identified with the kernel D m (H) of the Lie bracket [ , ] : H ⊗ Lie m+1 (H) → Lie m+2 (H).…”

Section: Introductionmentioning

confidence: 99%

Alternative versions of the Johnson homomorphisms and the LMO functor

Vera

2019

Preprint

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Let Σ be a compact connected oriented surface with one boundary component and let M denote the mapping class group of Σ. By considering the action of M on the fundamental group of Σ it is possible to define different filtrations of M together with some homomorphisms on each term of the filtration. The aim of this paper is twofold. Firstly we study a filtration of M introduced recently by Habiro and Massuyeau, whose definition involves a handlebody bounded by Σ. We shall call it the "alternative Johnson filtration", and the corresponding homomorphisms are referred to as "alternative Johnson homomorphisms". We provide a comparison between the alternative Johnson filtration and two previously known filtrations: the original Johnson filtration and the Johnson-Levine filtration. Secondly, we study the relationship between the alternative Johnson homomorphisms and the functorial extension of the Le-Murakami-Ohtsuki invariant of 3-manifolds. We prove that these homomorphisms can be read in the tree reduction of the LMO functor. In particular, this provides a new reading grid for the tree reduction of the LMO functor. Contents 1. Introduction 2. Spaces of Jacobi diagrams and their operations 2.1. Generalities 2.2. Operations on Jacobi diagrams 3. The Kontsevich integral and the LMO functor 3.1. Kontsevich Integral 3.2. The LMO functor 4. Johnson-type filtrations 4.1. Preliminaries 4.2. Alternative Torelli group 4.3. Alternative Johnson filtration 5. Johnson-type homomorphisms 5.1. Preliminaries 5.2. Alternative Johnson homomorphisms 5.3. Alternative Johnson homomorphism on L 5.4. Diagrammatic versions of the Johnson-type homomorphisms 6. Alternative Johnson homomorphisms and the LMO functor 6.1. The filtration on Lagrangian cobordisms induced by the alternative degree 6.2. First alternative Johnson homomorphism and the LMO functor 6.3. Higher alternative Johnson homomorphisms and the LMO functor References

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Abelian quotients of subgroups of the mapping class group of surfaces

Cited by 170 publications

References 14 publications

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Triviality of the $J_4$-equivalence among homology 3-spheres

Alternative versions of the Johnson homomorphisms and the LMO functor

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