Mapping class groups, characteristic classes, and Bernoulli numbers

Abstract

The mapping class group Γg of a closed, connected and oriented surface Sg of genus g is defined as the group of connected components of the group of orientation preserving diffeomorphisms of Sg. This group has been the object of many recent studies. Of particular interest are its finite subgroups; these are precisely the finite groups which occur as groups of symmetries of the surface Sg equipped with a complex structure (a Riemann surface). The interplay of algebra, topology and analysis in the study of Γg make it one of the most fascinating groups. As it is the case for classical arithmetic groups, the finite subgroups of Γg are related to certain concepts in number theory. We shall discuss in this essay invariants of Γg which are related to number theory via Bernoulli numbers. The invariants we have in mind are firstly certain characteristic classes, associated with a natural flat vector bundle over BΓg, secondly, the orbifold Euler characteristic of the group Γg and thirdly its Yagita invariant. The characteristic classes are related to the denominators of Bernoulli numbers, the Euler characteristic involves the whole Bernoulli numbers, and our theorems concerning the Yagita invariant have to do with the notion of regular primes, which is expressible in terms of numerators of Bernoulli numbers. Although the three concepts which we study seem rather unrelated from the point of view of their definitions, the fact that they all are tightly linked to properties of finite subgroups and their normalizers and centralizers in Γg renders it plausible, that the resulting invariants must be somehow linked. The precise relationship, however, remains for the time being a mystery. We have tried to make these notes easy to read for non-experts. We therefore recall in Sections 1 through 3 many facts and definitions, and state the relevant properties of the mapping class group without proofs, in form of a survey. We also have included a substantial bibliography, helping the reader to find the proofs of the basic theorems of the subject, which are scattered through the literature and which are crucial for analyzing homological properties of the mapping class group. For the classical part, not dealing with the cohomology of the mapping class group, the reader should consult Birman’s book [Bi]. Section 4 contains a short introduction to the theory of characteristic classes for group representations, and in Section 5 we