Abelian quotients of the $Y$-filtration on the homology cylinders via the LMO functor

Invariants and structures of the homology cobordism group of homology cylinders

The homology cobordism group of homology cylinders is a generalization of the mapping class group and the string link concordance group. We study this group and its filtrations by subgroups by developing new homomorphisms. First, we define extended Milnor invariants by combining the ideas of Milnor's link invariants and Johnson homomorphisms. They give rise to a descending filtration of the homology cobordism group of homology cylinders. We show that each successive quotient of the filtration is free abelian of finite rank. Second, we define Hirzebruch-type intersection form defect invariants …

Homology cobordism group of homology cylinders and invariants related to lower central series

The homology cobordism group of homology cylinders is a generalization of both the mapping class group of surfaces and the string link concordance group. We consider extensions of Johnson homomorphisms of a mapping class group, Milnor invariants and Orr invariants of links to homology cylinders, all of which are related to free nilpotent groups. We establish a combined filtration via kernels of extended Johnson homomorphisms and extended Milnor invariants. We determine its image under the three invariants, and investigate relations among the invariants, and relations of the filtration to autom…

Homology cylinders: an enlargement of the mapping class group

We consider a homological enlargement of the mapping class group, defined by homology cylinders over a closed oriented surface (up to homology cobordism). These are important model objects in the recent Goussarov-Habiro theory of finite-type invariants of 3-manifolds. We study the structure of this group from several directions: the relative weight filtration of Dennis Johnson, the finite-type filtration of Goussarov-Habiro, and the relation to string link concordance. We also consider a new Lagrangian filtration of both the mapping class group and the group of homology cylinders.

同调柱胚幺半群的 $Y$-滤过相关分次中的挠元素

Torsion elements in the associated graded of the $Y$-filtration of the monoid of homology cylinders

本文研究了同调柱胚幺半群 $Y$-滤过相关分次模的挠子模，引入由 LMO 函子诱导的同态来检测挠元素，并证明 $Y_6\mathcal{IC}/Y_7$ 中非平凡挠元素的阶均为 $3$。

同调柱的非交换Reidemeister-Turaev挠率

A non-commutative Reidemeister-Turaev torsion of homology cylinders

本研究计算了同调柱的非交换Reidemeister-Turaev挠率，证明其约化结果为d次有限型不变量，且该挠率首项可恢复LMO同态1-圈部分与Enomoto-Satoh迹，为低维拓扑不变量研究提供新工具。

Symplectic Jacobi diagrams and the Lie algebra of homology cylinders

Let S be a compact connected oriented surface, whose boundary is connected or empty. A homology cylinder over the surface S is a cobordism between S and itself, homologically equivalent to the cylinder over S. The Y-filtration on the monoid of homology cylinders over S is defined by clasper surgery. Using a functorial extension of the Le-Murakami-Ohtsuki invariant, we show that the graded Lie algebra associated to the Y-filtration is isomorphic to the Lie algebra of ``symplectic Jacobi diagrams.'' This Lie algebra consists of the primitive elements of a certain Hopf algebra whose multiplicatio…

meanders，超椭圆pillowcase覆盖，以及Johnson filtration

Meanders, hyperelliptic pillowcase covers, and the Johnson filtration

本文通过构造特定meanders和双曲面pillowcase覆盖，证明了双曲连通分量中存在任意深的Johnson filtration中的比优化伪Anosov映射。

Equivalence relations for homology cylinders and the core of the Casson invariant

Let R be a compact oriented surface of genus g with one boundary component. Homology cylinders over R form a monoid IC into which the Torelli group I of R embeds by the mapping cylinder construction. Two homology cylinders M and M' are said to be Y_k-equivalent if M' is obtained from M by "twisting" an arbitrary surface S in M with a homeomorphim belonging to the k-th term of the lower central series of the Torelli group of S. The J_k-equivalence relation on IC is defined in a similar way using the k-th term of the Johnson filtration. In this paper, we characterize the Y_3-equivalence with thr…

关于同调圆柱李代数挠元的Sp-结构

On the Sp-structure of the torsion of the Lie algebra of homology cylinders

本文研究同调圆柱李代数挠元的Sp-模结构，通过改进LMO函子模1约化与钳子演算工具，证明挠部分满射到2-挠Sp-模，并给出三次分量的内在描述，进而理解Torelli群挠现象。

Morita's trace maps on the group of homology cobordisms

Morita introduced in 2008 a 1-cocycle on the group of homology cobordisms of surfaces with values in an infinite-dimensional vector space. His 1-cocycle contains all the "traces" of Johnson homomorphisms which he introduced fifteen years earlier in his study of the mapping class group. In this paper, we propose a new version of Morita's 1-cocycle based on a simple and explicit construction. Our 1-cocycle is proved to satisfy several fundamental properties, including a connection with the Magnus representation and the LMO homomorphism. As an application, we show that the rational abelianization…

Finite type invariants and fatgraphs

We define an invariant $\nabla_G(M)$ of pairs M,G, where M is a 3-manifold obtained by surgery on some framed link in the cylinder $S\times I$, S is a connected surface with at least one boundary component, and G is a fatgraph spine of S. In effect, $\nabla_G$ is the composition with the $ι_n$ maps of Le-Murakami-Ohtsuki of the link invariant of Andersen-Mattes-Reshetikhin computed relative to choices determined by the fatgraph G; this provides a basic connection between 2d geometry and 3d quantum topology. For each fixed G, this invariant is shown to be universal for homology cylinders, i.e.,…

Tree homology and a conjecture of Levine

In his study of the group of homology cylinders, J. Levine made the conjecture that a certain homomorphism eta': T -> D' is an isomorphism. Here T is an abelian group on labeled oriented trees, and D' is the kernel of a bracketing map on a quasi-Lie algebra. Both T and D' have strong connections to a variety of topological settings, including the mapping class group, homology cylinders, finite type invariants, Whitney tower intersection theory, and the homology of the group of automorphisms of the free group. In this paper, we confirm Levine's conjecture. This is a central step in classifying …