Ho, Man-Ho Remarks on flat and differential \(K\)-theory. (Remarques sur les \(K\)-théories plate et différentielle.) (English) Zbl 1329.19010 Ann. Math. Blaise Pascal 21, No. 1, 91-101 (2014). [Note: Some diagrams below cannot be displayed correctly in the web version. Please use the PDF version for correct display.]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)]. Reviewer: Seunghun Hong (Orange City) Cited in 3 Documents MSC: 19L50 Twisted \(K\)-theory; differential \(K\)-theory 19K56 Index theory 58J22 Exotic index theories on manifolds Keywords:differential \(K\)-theory; flat \(K\)-theory; topological index Citations:Zbl 1202.19007; Zbl 1197.58007 PDFBibTeX XMLCite \textit{M.-H. Ho}, Ann. Math. Blaise Pascal 21, No. 1, 91--101 (2014; Zbl 1329.19010) Full Text: DOI arXiv References: [1] Baum, P.; Higson, N.; Schick, T., On the equivalence of geometric and analytic \(K\)-homology, Pure Appl. Math. Q., 3, 1-3, 1-24 (2007) · Zbl 1146.19004 · doi:10.4310/PAMQ.2007.v3.n1.a1 [2] Bismut, J. M.; Zhang, W., Real embeddings and eta invariants, Math. Ann., 295, 4, 661-684 (1993) · Zbl 0795.57010 · doi:10.1007/BF01444909 [3] Bunke, U., Index theory, eta forms, and Deligne cohomology, Mem. Amer. Math. Soc., 198, 928 (2009) · Zbl 1181.58017 [4] Bunke, U.; Schick, T., Smooth \(K\)-theory, Astérisque, 328, 45-135 (2009) · Zbl 1202.19007 [5] Bunke, U.; Schick, T., Uniqueness of smooth extensions of generalized cohomology theories, J. Topol., 3, 1, 110-156 (2010) · Zbl 1252.55002 · doi:10.1112/jtopol/jtq002 [6] Bunke, U.; Schick, T.; Bär, C.; Lohkamp, J.; Schwarz, M., Differential \(K\)-theory. A survey, Global Differential Geometry, 17, 303-358 (2012) · Zbl 1245.19002 [7] Freed, D.; Lott, J., An index theorem in differential \({K}\)-theory, Geom. Topol., 14, 2, 903-966 (2010) · Zbl 1197.58007 · doi:10.2140/gt.2010.14.903 [8] Ho, M.-H., The differential analytic index in Simons-Sullivan differential \(K\)-theory, Ann. Global Anal. Geom., 42, 4, 523-535 (2012) · Zbl 1257.19005 · doi:10.1007/s10455-012-9325-1 [9] Hopkins, M. J.; Singer, I. M., Quadratic functions in geometry, topology, and \(M\)-theory, J. Differential Geom., 70, 3, 329-452 (2005) · Zbl 1116.58018 [10] Klonoff, K., An index theorem in differential \(K\)-theory (2008) [11] Lott, J., \( \mathbb{R}/\mathbb{Z}\) index theory, Comm. Anal. Geom., 2, 2, 279-311 (1994) · Zbl 0840.58044 [12] Simons, J.; Sullivan, D., Structured vector bundles define differential \(K\)-theory, Quanta of maths, 11, 579-599 (2010) · Zbl 1216.19009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.