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Remarks on flat and differential \(K\)-theory. (Remarques sur les \(K\)-théories plate et différentielle.) (English) Zbl 1329.19010

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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):
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: \[ \begin{tikzcd} K^{-1}(X;\mathbb{R}/\mathbb{Z}) \ar[r, "i"]\ar[d,"{\mathrm{ind}^{\mathrm{top}}}" '] &\hat K^0 (X) \ar[d,"{\mathrm{ind}^{\mathrm{top}}}"]\\ K^{-1}(B;\mathbb{R}/\mathbb{Z}) \ar[r, "i" '] &\hat K^0 (B) \end{tikzcd} \] Here, the horizontal maps are the canonical inclusion maps.
2.
The existence of the unique natural isomorphism between the differential \(K\)-group of U. Bunke and T. Schick [Astérisque 328, 45–135 (2009; Zbl 1202.19007)] and that of D. S. Freed and J. Lott [Geom. Topol. 14, No. 2, 903–966 (2010; Zbl 1197.58007)].

MSC:

19L50 Twisted \(K\)-theory; differential \(K\)-theory
19K56 Index theory
58J22 Exotic index theories on manifolds
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References:

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