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 nontriviality 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.

No abstract

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 in [12] and [13], 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.

ABSTRACT. We study the structure of the symplectic invariant part h Sp g,1 of the Lie algebra h g,1 consisting of symplectic derivations of the free Lie algebra generated by the rational homology group of a closed oriented surface Σ 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 Sp-invariant component of the bracket of two elements in h g,1 . Second we clarify the relations among h g,1 and the other two related Lie algebras h g, * and h g which correspond to the cases of a closed surface Σ g with and without base point * ∈ Σ 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 h Sp g,1 , we reveal a considerable difference between two particular submodules of it, one is the Sp-invariant part of a certain ideal j g,1 and the other is that of the Johnson image.Finally we combine these results to determine the structure of 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.

803The Magnus representation and higher-order Alexander invariants for homology cobordisms of surfaces TAKUYA SAKASAIThe 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 and Wang's argument over the Gassner representation of string links. Then, by applying Cochran and Harvey's framework of higher-order (noncommutative) Alexander invariants to them, we extract several information about the monoid and related objects.57M05; 57M27, 20F34, 57N05 IntroductionLet † g;1 be a compact connected oriented surface of genus g 1 with one boundary component. A homology cylinder over † g;1 consists of a homology cobordism from † g;1 to itself with markings of its boundary. We denote by C g;1 the set of isomorphisms classes of homology cylinders. Since stacking two homology cylinders gives a new one, we can endow C g;1 with a monoid structure (see Section 2 for the precise definition). The origin of homology cylinders goes back to Goussarov [10], Habiro [11], Garoufalidis-Levine [9] and Levine [21], where the clasper (or clover) surgery theory is effectively used to investigate the structure of C g;1 .As mentioned in [9] and [21], we can construct a homology cylinder from each homology 3-sphere or pure string link. Also, for a given homology cylinder, we can use an element of the mapping class group M g;1 of † g;1 to give another one by changing its markings. Since these operations preserve each monoid structure, C g;1 can be regarded as a simultaneous generalization of the monoid of homology 3-spheres, that of string links and M g;1 , any of which plays an important role in the theory of 3-manifolds.On the other hand, there exist some natural ways (see Section 6.3) to construct closed 3-manifolds from each homology cylinder. Therefore, by using its monoid structure, C g;1 serves as a tool for classifying closed 3-manifolds.The aim of this paper is to study the structure of C g;1 from an algebraic point of view. The main tools for that are invariants using noncommutative rings arising from group rings of fundamental groups of homology cylinders. The first half of this paper is occupied by defining the Magnus representation for homology cylinders and observing its fundamental properties. In the latter half, Cochran and Harvey's higherorder Alexander invariants are introduced and used to define numerical invariants for matrices obtained by the Magnus representation or associated Reidemeister torsions, whose entries are in noncommutative rings so that it is difficult to treat directly. This combination of the Magnus representation and higher-order Alexander invariants leads us to derive various facts about the structure of C g;1 and to give applications to the theory of closed 3-manifolds.The outline of this paper is as follows.In Section 2, we first recall the definition of homology cylinders over † g;1 by...