Remarks on flat and differential \(K\)-theory
type: Item
instance of: scholarly article
title: Remarks on flat and differential \(K\)-theory
zbMATH Open document ID: 1329.19010
author: Man-Ho Ho
published in: Annales Mathématiques Blaise Pascal
publication date: 2014-08-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1203.5383
reviewed by: Seunghun Hong
Mathematics Subject Classification ID: 19L50
Mathematics Subject Classification ID: 19K56
Mathematics Subject Classification ID: 58J22
zbMATH DE Number: 6329058
zbMATH Keywords: differential \(K\)-theory
zbMATH Keywords: flat \(K\)-theory
zbMATH Keywords: topological index
MaRDI profile type: Publication
OpenAlex ID: W2963323679
arXiv ID: 1203.5383
cites work: On the equivalence of geometric and analytic \(K\)-homology
cites work: Real embeddings and eta invariants
cites work: Index theory, eta forms, and Deligne cohomology
cites work: Smooth K-Theory
cites work: Uniqueness of smooth extensions of generalized cohomology theories
cites work: Differential K-Theory: A Survey
cites work: An index theorem in differential \(K\)-theory
cites work: The differential analytic index in Simons-Sullivan differential \(K\)-theory
cites work: Quadratic functions in geometry, topology, and \(M\)-theory
cites work: \(\mathbb{R} /\mathbb{Z}\) index theory
cites work: Structured vector bundles define differential K-theory
DOI: 10.5802/AMBP.337
review text: 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}}
description: scientific article; zbMATH DE number 6329058