Let $\mathcal{M}$ denote the mapping class group of Σ, a compact connected oriented surface with one boundary component. The action of $\mathcal{M}$ on the nilpotent quotients of π1(Σ) allows to define the so-called Johnson filtration and the Johnson homomorphisms. J. Levine introduced a new filtration of $\mathcal{M}$, called the Lagrangian filtration. He also introduced a version of the Johnson homomorphisms for this new filtration. The first term of the Lagrangian filtration is the Lagrangian mapping class group, whose definition involves a handlebody bounded by Σ, and which contains the Torelli group. These constructions extend in a natural way to the monoid of homology cobordisms. Besides, D. Cheptea, K. Habiro and G. Massuyeau constructed a functorial extension of the LMO invariant, called the LMO functor, which takes values in a category of diagrams. In this paper we give a topological interpretation of the upper part of the tree reduction of the LMO functor in terms of the homomorphisms defined by J. Levine for the Lagrangian mapping class group. We also compare the Johnson filtration with the filtration introduced by J. Levine.

中文翻译：

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Johnson-Levine 同态和 LMO 函子的树归约

让$\数学{M}$表示 Σ 的映射类组，一个具有一个边界分量的紧凑连通定向表面。的行动$\数学{M}$关于幂零商π1(Σ) 允许定义所谓的约翰逊过滤和约翰逊同态。J. Levine 介绍了一种新的过滤$\数学{M}$， 叫做拉格朗日过滤. 他还为这种新过滤引入了约翰逊同态的一个版本。拉格朗日过滤的第一项是拉格朗日映射类群，其定义涉及以 Σ 为界的手柄体，其中包含 Torelli 群。这些构造以一种自然的方式延伸到同源协边的幺半群。此外，D. Cheptea、K. Habiro 和 G. Massuyeau 构建了 LMO 不变量的函子扩展，称为 LMO 函子，它采用图表类别中的值。在本文中，我们给出了拓扑解释上面的部分根据 J. Levine 为拉格朗日映射类组定义的同态性，LMO 函子的树归约。我们还将 Johnson 过滤与 J. Levine 介绍的过滤进行了比较。