Wall's obstructions and Whitehead torsion (Q792661)

For a finitely dominated CW complex X let w(X)\(\in \tilde K_ 0({\mathbb{Z}}\pi_ 1(X))\) denote Wall's finiteness obstruction and let \(\sigma(X)\in Wh(X\times S^ 1)\) be the finiteness obstruction defined by \textit{S. Ferry} [Lect. Notes Math. 870, 73-81 (1981; Zbl 0472.57014)]. The author proves that (identifying \(\tilde K_ 0({\mathbb{Z}}\pi_ 1(X))\) with \(\tilde K_ 0(X)\) as usual) \(\sigma\) (X) maps to w(X) under the Bass-Heller-Swan projection \(Wh(X\times S^ 1)\to \tilde K_ 0(X)\). In a note added in proof the author claims that results of \textit{T. A. Chapman} [Trans. Am. Math. Soc. 262, 303-334 (1980; Zbl 0464.57009)] imply that actually, \(\sigma\) (X) belongs to the summand \(\tilde K_ 0(X)\) in the Bass-Heller-Swan decomposition of \(Wh(X\times S^ 1)\). This claim is not correct as it stands. In fact if it were true then Bass, Heller and Swan's injection \(h: \tilde K_ 0({\mathbb{Z}}\pi_ 1X)\to Wh(\pi_ 1X\times {\mathbb{Z}})\) would map w(X) to \(\sigma\) (X). However, S. Ferry [loc. cit.] shows that \(\sigma\) (X) is invariant under transfer maps associated to finite covers \(X\times S^ 1\to X\times S^ 1\). And it is easy to see that Im(h) is not invariant under such transfers. It follows from work of \textit{A. A. Ranicki} [unpublished preprint; see also Abstr. Pap. Am. Math. Soc. 3 (1982), no. 793-18-8] that w(X) maps to \(\sigma\) (X) under a different injection \(h': \tilde K_ 0({\mathbb{Z}}\pi_ 1X)\to Wh(\pi_ 1X\times {\mathbb{Z}}).\) If P is a f.g. projective \({\mathbb{Z}}\pi_ 1X\)-module and \(Q=P\otimes_{{\mathbb{Z}}\pi_ 1X}{\mathbb{Z}}[\pi_ 1X\times {\mathbb{Z}}]\) then \(h([P])=[Q;z1_ Q]\) while \(h'([P])=[Q;-z1_ Q]\). The author's mistake is not original. In fact there seems to be several papers in the literature where it is implicitly or explicitly assumed that a given geometrically defined injection \(\tilde K_ 0({\mathbb{Z}}\pi)\to Wh(\pi \times {\mathbb{Z}})\) coincides with h.