On Wh2 of a Bieberbach group

Abstract

The groups which occur as the fundamental group of a compact flat Riemannian manifold are called Bieberbach groups[21]. Bieberbach groups are characterized algebraically as the torsion free groups which contain a unique maximal normal finitely generated abelian subgroup of finite index. If r is a Bieberbach group and A c r its unique maximal normal abelian subgroup of finite index then T/A is called the holonomy group of r and the rank of r is defined to be the rank of A. A homomorphism fi r +T is said to be s-expansive, where s is an integer, if its restriction to A is multiplication by s and it induces the identity map on T/A. Recall that a monomorphism of groups f: n-n’ induces a transfer map f*: Whi(17’)+ Whi(17) i = 0, 1, 2 provided the image off has finite index in l7’ (see [13] for the case i = 2). If B is an abelian group then the notation Bodd will be used for B @Z[1/2]. The following proposition, which is proved by geometric methods in the next section, is crucial to the proof of the main theorem: