An elementary differential extension of odd \(K\)-theory. (Q2874241)

The main result of the paper under review states that the space of smooth mappings \(M\to U(\infty)\), under a suitable equivalence relation, is a differential refinement \(\hat{K}^{-1}(M)\) of odd differential \(K\)-theory on the manifold \(M\). Differential \(K\)-theory is a refinement of \(K\)-theory, containing additional information at the level of differential forms. Such refinements have been studied since work of \textit{J. Cheeger} and \textit{J. Simons} [Lect. Notes Math. 1167, 50--80 (1985; Zbl 0621.57010)], and in later years the work by \textit{U. Bunke} and \textit{T. Schick} [in: From probability to geometry II. Volume in honor of the 60th birthday of Jean-Michel Bismut. Paris: Société Mathématique de France (SMF). 45--135 (2009; Zbl 1202.19007)] gives a more axiomatic approach. The paper provides a thorough introduction to Chern-Simons forms before proving the main results. The cocycles in the model are simple and the curvature and the mapping \(I:\hat{K}^{-1}(M)\to K^{-1}(M)\) are explicitly defined at the level of cocycles. However, the mapping \(a:\Omega^{\text{ev}}(M)/\mathrm{Im}(\mathrm{d})\to \hat{K}^{-1}(M)\) is only defined by means of abstractly inverting a certain isomorphism. This is remedied in the appendix by adding further cocycles admitting an explicit \(a\)-mapping.