On the Whitehead group of Novikov rings associated to irrational homomorphisms
Given a homomorphism \(\xi\colon G\to\mathbb R\), the Novikov ring \(\widehat{\mathbb Z G_\xi}\) is a completion of the group ring \(\mathbb Z G\). The main result of this paper is that the natural homomorphism \(\text{Wh}(G)\to\text{Wh}(G;\xi)\) from the Whitehead group of \(G\) to the Whitehead group of the Novikov ring is surjective. The group \(\text{Wh}(G;\xi)\) was introduced by \textit{F. Latour} [``Existence de 1-formes fermées non singulières dans une classe de cohomologie de de Rham'', Publ. Math., Inst. Hautes Étud. Sci. 80, 135--194 (1994; Zbl 0837.58002)] as the receptacle for an obstruction for the existence of a nonsingular closed \(1\)-form in the cohomology class of \(\xi\colon G=\pi_1M\to\mathbb R\) of a closed, connected smooth manifold \(M\) with \(\dim M\geq 6\). The proof of the main result builds on previous work of \textit{A. V. Pajitnov} and \textit{A. A. Ranicki} [``The Whitehead group of the Novikov ring'', \(K\)-Theory 21, 325--365 (2000; Zbl 0996.19002)], where, in particular, the main result is established in the case that \(\xi\) factors through the integers -- hence, the ``irrational homomorphisms'' of the title. The algebraic proof yields results for the Novikov ring \(\widehat{RG_\xi}\), where \(R\) is any ring with unit.
scientific article; zbMATH DE number 5121386
scientific article