Remarks on flat and differential \(K\)-theory
The main purpose of this paper is to explicitly write down the proofs of the following two facts (which seem to have been known to the experts in the field but missing in the literature): {\parindent=6mm \begin{itemize}\item[1.] The compatibility of the topological index maps in the flat and differential \(K\)-theories. In other words, for a smooth fiber bundle \(X\to B\) where \(X\) is compact, the following diagram commutes: NEWLINE\[CARRIAGE_RETURNNEWLINE\begin{tikzcd} K^{-1}(X;\mathbb{R}/\mathbb{Z}) \ar[r, "i"]\ar[d,"{\mathrm{ind}^{\mathrm{top}}}" '] &\hat K^0 (X) \ar[d,"{\mathrm{ind}^{\mathrm{top}}}"]\\ K^{-1}(B;\mathbb{R}/\mathbb{Z}) \ar[r, "i" '] &\hat K^0 (B) \end{tikzcd}CARRIAGE_RETURNNEWLINE\]NEWLINE Here, the horizontal maps are the canonical inclusion maps. \item[2.] The existence of the unique natural isomorphism between the differential \(K\)-group of \textit{U. Bunke} and \textit{T. Schick} [Astérisque 328, 45--135 (2009; Zbl 1202.19007)] and that of \textit{D. S. Freed} and \textit{J. Lott} [Geom. Topol. 14, No. 2, 903--966 (2010; Zbl 1197.58007)]. NEWLINENEWLINE\end{itemize}}
scientific article; zbMATH DE number 6329058
scientific article