Shigeyuki Morita

Generators for the tautological algebra of the moduli space of curves

Abstract In this paper, we prove that the tautological algebra in cohomology of the moduli space Mg of smooth projective curves of genus g is generated by the first [g/3] Mumford–Morita–Miller classes. This solves a part of Fabers conjecture (Moduli of Curves and Abelian Varieties Vieweg, Braunschweig, 1999) concerning the structure of the tautological algebra affirmatively. More precisely, for any k we express the kth Mumford–Morita–Miller class ek as an explicit polynomial in the lower classes for all genera g=3k−1, 3k−2,…,2 .

Characteristic classes of surface bundles

Let M be a C~-manifold. If one wants to classify all differentiable fibre bundles over a given manifold which have M as fibres, then one must study the topology of the corresponding structure group DiffM, the group of all diffeomorphisms of M equipped with the C ~ topology. It is easy to see that the connected component Diffo M of the identity of Diff M is an open subgroup. Therefore the natural topology on the quotient group @(M)= Diff M/Diffo M, the diffeotopy group of M, is the discrete topology. Hence we have a fibration

Signatures of foliated surface bundles and the symplectomorphism groups of surfaces

Abstract For any closed oriented surface Σ g of genus g ⩾3, we prove the existence of foliated Σ g -bundles over surfaces such that the signatures of the total spaces are non-zero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form ω on the fiber. We relate the cohomology class represented by the transverse symplectic form to a crossed homomorphism Flux : Symp Σ g →H 1 (Σ g ; R ) which is an extension of the flux homomorphism Flux : Symp 0 Σ g →H 1 (Σ g ; R ) from the identity component Symp 0 Σ g to the whole group Symp Σ g of symplectomorphisms of Σ g with respect to the symplectic form ω .

Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves

Infinite presentations are given for all of the higher Torelli groups of once-punctured surfaces. In the case of the classical Torelli group, a finite presentation of the corresponding groupoid is also given, and finite presentations of the classical Torelli groups acting trivially on homology modulo N are derived for all N. Furthermore, the first Johnson homomorphism, which is defined from the classical Torelli group to the third exterior power of the homology of the surface, is shown to lift to an explicit canonical 1-cocycle of the Teichmuller space. The main tool for these results is the known mapping class group invariant ideal cell decomposition of the Teichmuller space.

Computations in formal symplectic geometry and characteristic classes of moduli spaces

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.

Crossed flux homomorphisms and vanishing theorems for Flux groups

Abstract.We study the flux homomorphism for closed forms of arbitrary degree, with special emphasis on volume forms and on symplectic forms. The volume flux group is an invariant of the underlying manifold, whose non-vanishing implies that the manifold resembles one with a circle action with homologically essential orbits.

Abelianizations of derivation Lie algebras of the free associative algebra and the free Lie algebra

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.

ON THE COBORDISM CLASSES OF CODIMENSION ONE FOLIATIONS WHICH ARE ALMOST WITHOUT HOLONOMY

INTRODUCYrION LET M be an oriented closed n-dimensional P-manifold and (M, 9) a transversely oriented codimension one P-foliation of M. The purpose of this paper is to study foliated cobordism class of (M, S) assuming that 9 is almost without holonomy. In virtue of the works of Haefliger [3,4], Mather [ 12,131 and Thurston [24,25], foliated cobordism groups can be studied in the following lines. Namely there is a universal space Brim, called the Haefliger’s classifying space for r,m-structures, so that any (M,

Integral Euler Characteristic of Out F11

r) determines an ndimensional’ homology class of Brim, which turns out to be closely related to the cobordism class of (M, 9). On the other hand there is a map BDiffKmlW -+ QBr,m from the classifying space of the discrete group DiffKm[W of all Cm-diffeomorphisms of Iw with compact support to the loop space RBr,m of sr,m, which induces an isomorphism on integral homology. Thus in some sense the study of homology classes of Brim can be reduced to the study of those of DiffKaIw. In our case, these two fundamental results work very well. Our main result is as follows. Let.(M, 9) be as before and assume that 9 is almost without holonomy. Then it is homologous to a disjoint union of finite number of foliated S-bundles over (n I)-dimensional tori. For n = 3, in particular, it follows that (M, 9) is foliated cobordant to a disjoint union of foliated S-bundles over T2. The foliated cobordism classes of foliated S’-bundles over T2 were studied by Tsuboi in [26]. Fukui and Oshikiri proved the nullity of the foliated cobordism classes of certain foliated 3-manifolds ([2,21]). By our method, we can give a wider class of foliated 3-manifolds which are foliated null-cobordant (Corollary to Theorem 2). Also together with results of Wallet [29] and Herman [7], we re-obtain the vanishing of the Godbillon-Vey class of an almost without holonomy foliation (M, 5) which we previously proved in [16]. The main tool of this paper is the notion of foliated J-bundles which we developed in [16] in order to calculate the Godbillon-Vey class. Associated to each (M, S), there is a foliated J-bundle and the original 9 can be “embedded” in it as the graph of 9. Then we can deform the underlying r,” -structure of 9 by simply moving this graph on the total space of the J-bundle. This method is originally due to Haefliger. The foliated J-bundles associated to (M, 9’) are determined by the holonomies of the compact leaves and the Novikov transformations (which depend on the non-compact leaves). These two data are essential. In fact, the structure of the foliated S’-bundles over T”-’ to which (M, 9:) is homologous, are determined by these data. Some of the results in this paper were contained in

Lie algebras of symplectic derivations and cycles on the moduli spaces

5 of our preprint [15] some part of which has been published in [ 161. We would like to refer the reader to [16] for the generalities of the foliation which are almost without holonomy and the construction of the associated foliated J-bundles and other related notions. In this paper, all manifolds, foliations and diffeomorphisms are assumed to be smooth (C=). Moreover. foliations will mean transversely oriented codimension one foliations.