On the higher Whitehead groups of a Bieberbach group

Let Γ\Gamma be a Bieberbach group, i.e. the fundamental group of a compact flat Riemannian manifold. In this paper we show that if p>2p > 2 is a prime, then the pp-torsion subgroup of Whi(Γ){\text {Wh}_i}(\Gamma ) vanishes for 0≤i≤2p−20 \leq i \leq 2p - 2, where Whi(Γ){\text {Wh}_i}(\Gamma ) is the iith higher Whitehead group of Γ\Gamma. The proof involves Farrell and Hsiang’s structure theorem for Bieberbach groups, parametrized surgery, pseudoisotopy, and Waldhausen’s algebraic KK-theory of spaces.