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In the late 1980's, it was shown that the Casson invariant appears in the difference between the two filtrations of the Torelli group: the lower central series and the Johnson filtration, and that its core part was identified with the secondary characteristic class $d_1$ associated with the fact that the first $\mathrm{MMM}$ class vanishes on the Torelli group (however it turned out that Johnson proved the former part highly likely prior to the above, see Remark 1.1). This secondary class $d_1$ is a rational generator of $H^1(\mathcal{K}_g;\mathbb{Z})^{\mathcal{M}_g}\cong\mathbb{Z}$ where $\mathcal{K}_g$ denotes the Johnson subgroup of the mapping class group $\mathcal{M}_g$. Hain proved, as a particular case of his fundamental result, that this is the only difference in degree $2$. In this paper, we prove that no other invariant than the above gives rise to new rational difference between the two filtrations up to degree $6$. We apply this to determine $H_1(\mathcal{K}_g;\mathbb{Q})$ explicitly by computing the description given by Dimca, Hain and Papadima. We also show that any finite type rational invariant of homology $3$-spheres of degrees up to $6$, including the second and the third Ohtsuki invariants, can be expressed by $d_1$ and lifts of Johnson homomorphisms., Comment: 23 pages, final version, to appear in Quantum Topology

We construct an abelian quotient of the symplectic derivation Lie algebra $\mathfrak{h}_{g,1}$ of the free Lie algebra generated by the fundamental representation of $\mathrm{Sp}(2g,\mathbb{Q})$. More specifically, we show that the weight $12$ part of the abelianization of $\mathfrak{h}_{g,1}$ is $1$-dimensional for $g \ge 8$. The computation is done with the aid of computers., Comment: 20 pages

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 of the group of homology cobordisms is non-trivial. Besides, we apply some of our algebraic methods to compare two natural filtrations on the automorphism group of a finitely-generated free group., Comment: 35 pages; minor changes

By analyzing how the Borel regulator classes vanish on various groups related to $\mathrm{GL}(n,\mathrm{Z})$, we define three series of secondary characteristic classes for subgroups of automorphism groups of free groups. The first case is the $\mathrm{IA}$-automorphism groups and we show that our classes coincide with higher $\mathrm{FR}$ torsions due to Igusa. The second case is the mapping class groups and our classes also turn out to be his higher torsions which are non-zero multiples of the Mumford-Morita-Miller classes of even indices. Our construction gives new group cocycles for these still mysterious classes. The third case is the outer automorphism groups of free groups of specific ranks. Here we give a conjectural geometric meaning to a series of unstable homology classes called the Morita classes. We expect that certain unstable secondary classes would detect them., Comment: 24pages, added references

We show that the integral Euler characteristic of the outer automorphism group of the free group of rank 11 is -1202., Comment: 7 pages, final version, to appear in Exp. Math

We study the structure of the symplectic invariant part $\mathfrak{h}_{g,1}^{\mathrm{Sp}}$ of the Lie algebra $\mathfrak{h}_{g,1}$ consisting of symplectic derivations of the free Lie algebra generated by the rational homology group of a closed oriented surface $\Sigma_{g}$ of genus $g$. First we describe the orthogonal direct sum decomposition of this space which is induced by the canonical metric on it and compute it explicitly up to degree $20$. In this framework, we give a general constraint which is imposed on the $\mathrm{Sp}$-invariant component of the bracket of two elements in $\mathfrak{h}_{g,1}$. Second we clarify the relations among $\mathfrak{h}_{g,1}$ and the other two related Lie algebras $\mathfrak{h}_{g,*}$ and $\mathfrak{h}_{g}$ which correspond to the cases of a closed surface $\Sigma_g$ with and without base point $*\in\Sigma_g$. In particular, based on a theorem of Labute, we formulate a method of determining these differences and describe them explicitly up to degree $20$. Third, by giving a general method of constructing elements of $\mathfrak{h}_{g,1}^{\mathrm{Sp}}$, we reveal a considerable difference between the two submodules of it, one is the $\mathrm{Sp}$-invariant part of a certain ideal $\mathfrak{j}_{g,1}$ and the other is that of the Johnson image. Finally we combine these results to determine the structure of $\mathfrak{h}_{g,1}$ completely up to degree $6$ including the unstable cases where the genus $1$ case has an independent meaning. In particular, we see a glimpse of the Galois obstructions explicitly from our point of view., Comment: 39 pages, 2 figures, final version

We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights.From these, we obtain some non-triviality results in each case. In particular, we determine the integral Euler characteristics of the outer automorphism groups Out F_n of free groups for all n <= 10 and prove the existence of plenty of rational cohomology classes of odd degrees. We also clarify the relationship of the commutative graph homology with finite type invariants of homology 3-spheres as well as the leaf cohomology classes for transversely symplectic foliations. Furthermore we prove the existence of several new non-trivalent graph homology classes of odd degrees. Based on these computations, we propose a few conjectures and problems on the graph homology and the characteristic classes of the moduli spaces of graphs as well as curves., Comment: 33 pages, final version, to appear in Quantum Topology

We show that a certain symmetry exists in the stable irreducible decomposition of the Lie algebra consisting of symplectic derivations of the free Lie algebra generated by the first homology group of compact oriented surfaces., Comment: 8 pages. Final version. To appear in RIMS K\^oky\^uroku Bessatsu

A homology cylinder over a compact manifold is a homology cobordism between two copies of the manifold together with a boundary parametrization. We study abelian quotients of the homology cobordism group of homology cylinders. For homology cylinders over general surfaces, it was shown by Cha, Friedl and Kim that their homology cobordism groups have infinitely generated abelian quotient groups by using Reidemeister torsion invariants. In this paper, we first investigate their abelian quotients again by using another invariant called the Magnus representation. After that, we apply the machinery obtained from the Magnus representation to higher dimensional cases and show that the homology cobordism groups of homology cylinders over a certain series of manifolds regarded as a generalization of surfaces have big abelian quotients. In the proof, a homological localization, called the acyclic closure, of a free group and its automorphism group play important roles and our result also provides some information on these groups from a group-theoretical point of view., Comment: 21 pages, 2 figures, results on the algebraic closure of a free group are added, final version, to appear in Journal of Mathematical Sciences, the University of Tokyo

We determine the abelianizations of the following three kinds of graded Lie algebras in certain stable ranges: derivations of the free associative algebra, derivations of the free Lie algebra and symplectic derivations of the free associative algebra. In each case, we consider both the whole derivation Lie algebra and its ideal consisting of derivations with positive degrees. As an application of the last case, and by making use of a theorem of Kontsevich, we obtain a new proof of the vanishing theorem of Harer concerning the top rational cohomology group of the mapping class group with respect to its virtual cohomological dimension., Comment: 30 pages, 18 figures. Title modified, final version, to appear in Duke Math. J

This is a survey of Magnus representations with particular emphasis on their applications to mapping class groups and monoids (groups) of homology cobordisms of surfaces. In the first half, we begin by recalling the basics of the Fox calculus and overview Magnus representations for automorphism groups of free groups and mapping class groups of surfaces with related topics. In the latter half, we discuss in detail how the theory in the first half extends to homology cobordisms of surfaces and present a number of applications from recent researches., Comment: 69 pages. The statement of Proposition 6.1 is corrected. Submitted to "Handbook of Teichmu"ller theory, volume III" (editor: A. Papadopoulos)

Homologically fibered knots are knots whose exteriors satisfy the same homological conditions as fibered knots. In our previous paper, we observed that for such a knot, higher-order Alexander invariants defined by Cochran, Harvey and Friedl are generally factorized into the part of the Magnus matrix and that of a certain Reidemeister torsion, both of which are known as invariants of homology cylinders over a surface. In this paper, we study more details of the invariants and give some concrete calculations by restricting to the case of the invariants associated with metabelian quotients of their knot groups. We provide examples of explicit calculations of the invariants for all the 12 crossings non-fibered homologically fibered knots., Comment: 25 pages, 18 figures, to appear in Journal of Knot Theory and Its Ramifications

We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play important roles in the interaction between the mapping class group and finite-type invariants of 3-manifolds. In this paper, we discuss these groups from a group (co)homological point of view. The results include the determination of their abelianizations, lower bounds of the second homology and remarks on the (co)homology of higher degrees. As a by-product of this investigation, we determine the second homology of the mapping class group of a surface of genus 3., Comment: 20 pages. The proof of Lemma 4.2 is corrected. To appear in Algebraic & Geometric Topology

A homology cylinder over a surface consists of a homology cobordism between two copies of the surface and markings of its boundary. The set of isomorphism classes of homology cylinders over a fixed surface has a natural monoid structure and it is known that this monoid can be seen as an enlargement of the mapping class group of the surface. We now focus on abelian quotients of this monoid. We show that both the monoid of all homology cylinders and that of irreducible homology cylinders are not finitely generated and moreover they have big abelian quotients. These properties contrast with the fact that the mapping class group is perfect in general. The proof is given by applying sutured Floer homology theory to homologically fibered knots studied in a previous paper., Comment: 10 pages, 1 figure, to appear in Geometriae Dedicata

Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured manifolds associated with a special class of knots which we call {\it homologically fibered knots}. Then we use invariants of homology cylinders to give applications to knot theory such as fibering obstructions, Reidemeister torsions and handle numbers of homologically fibered knots., Comment: 24 pages, 9 figures, The title has been changed

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., Comment: 24 pages. An insufficient point in the original argument given in Section 4.2 is corrected by the referee's comment. To appear in Math. Proc. Cambridge Philos. Soc

We give a Dehn-Nielsen type theorem for the homology cobordism group of homology cylinders by considering its action on the acyclic closure, which was defined by Levine, of a free group. Then we construct an additive invariant of those homology cylinders which act on the acyclic closure trivially. We also describe some tools to study the automorphism group of the acyclic closure of a free group generalizing those for the automorphism group of a free group or the homology cobordism group of homology cylinders., Comment: This is the version published by Algebraic & Geometric Topology on 7 April 2006

The set of homology cobordisms from a surface to itself with markings of their boundaries has a natural monoid structure. To investigate the structure of this monoid, we define and study its Magnus representation and Reidemeister torsion invariants by generalizing Kirk-Livingston-Wang's argument over the Gassner representation of string links. Moreover, by applying Cochran and Harvey's framework of higher-order (non-commutative) Alexander invariants to them, we extract several pieces of information about the monoid and related objects., Comment: 28 pages. The whole paper has been rewritten, and the title has been changed