On \(Wh_ 2\) of a Bieberbach group
It is shown that for a Bieberbach group \(\Gamma\), the second Whitehead group \(Wh_ 2(\Gamma)\) is trivial, at least if 2-torsion is ignored. The special role of 2-torsion is explained by the fact that the author needs to represent elements of \(Wh_ 2(\Gamma)\) geometrically in terms of block homeomorphisms (not just pseudo-isotopies) - this works only away from 2; using such geometric representation the author shows: Let \(b\in Wh_ 2(\Gamma)_{odd}\). Then there is an integer \(N_ b\) such that if \(f: \Gamma\) \(\to \Gamma\) is ''s-expansive'' and \(s>N_ b\) then \(f^*(b)=0\) in \(Wh_ 2(\Gamma)_{odd}\) (where \(f^*\) denotes the transfer induced from the monomorphism f). This result and others are then used to apply Frobenius induction in order to prove the vanishing theorem.
scientific article; zbMATH DE number 3947234
scientific article