Toshiyuki Akita

Homological infiniteness of Torelli groups

Abstract The rational homology of the Torelli group of genus g relative to n distinguished points and r fixed embedded disks is proved to be infinite dimensional if g is sufficiently large relative to n+r. In particular, the rational homology of the (classical) Torelli group of genus g is infinite dimensional when g⩾7. In addition, the rational homology of the subgroup of the Torelli group of genus g generated by all the Dehn twists along separating simple closed curves is proved to be infinite dimensional when g⩾2.

Periodic surface automorphisms and algebraic independence of Morita–Mumford classes

Abstract We prove a vanishing theorem for the Morita–Mumford classes on periodic surface automorphisms, and construct enough periodic automorphisms to give an alternative and elementary proof of the stable rational algebraic independence of the Morita–Mumford classes, originally shown by Miller (J. Differential Geom. 24 (1986) 1–14) and Morita (Invent. Math. 90 (1987) 551–557).

Euler Characteristics of Coxeter Groups, PL-Triangulations of Closed Manifolds, and Cohomology of Subgroups of Artin Groups

The motivation for the theory of Euler characteristics of groups, which was introduced by C. T. C. Wall [ 21 ], was topology, but it has interesting connections to other branches of mathematics such as group theory and number theory. This paper investigates Euler characteristics of Coxeter groups and their applications. In his paper [ 20 ], J.-P. Serre obtained several fundamental results concerning the Euler characteristics of Coxeter groups. In particular, he obtained a recursive formula for the Euler characteristic of a Coxeter group, as well as its relation to the Poincare series (see §3). Later, I. M. Chiswell obtained in [ 10 ] a formula expressing the Euler characteristic of a Coxeter group in terms of orders of finite parabolic subgroups (Theorem 1). These formulae enable us to compute Euler characteristics of arbitrary Coxeter groups. On the other hand, the Euler characteristics of Coxeter groups W happen to be intimately related to their associated complexes [Fscr ] W , which are defined by means of the posets of nontrivial parabolic subgroups of finite order (see §2.1 for the precise definition). In particular, it follows from the recent result of M. W. Davis [ 13 ] that if [Fscr ] W is a product of a simplex and a generalized homology 2 n -sphere, then the Euler characteristic of W is zero (Corollary 3.1). The first objective of this paper is to generalize the previously mentioned result to the case when [Fscr ] W is a PL-triangulation of a closed 2 n -manifold which is not necessarily a homology 2 n -sphere. In other words (as given below in Theorem 3), if W is a Coxeter group such that [Fscr ] W is a PL- triangulation of a closed 2 n -manifold, then the Euler characteristic of W is equal to 1−χ([Fscr ] W )/2.

A vanishing theorem for the p-local homology of Coxeter groups

Given an odd prime number

A formula for the Euler characteristics of even dimensional triangulated manifolds

p

Second mod 2 homology of Artin groups

and a Coxeter group

Integral Riemann–Roch formulae for cyclic subgroups of mapping class groups

W

Aspherical Coxeter Groups That are Quillen Groups

such that the order of the product

Vanishing ranges for the mod

st

p

is prime to