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The index map in algebraic \(K\)-theory. (English) Zbl 1391.19005

In [Prog. Math. 244, 263–304 (2006; Zbl 1108.14012)], V. Drinfeld set problems: (A) Show the two notions of \(K\)-theory torsor mentioned in [loc .cit.] coincide. (B) Associate to a Tate \(R\)-module a \(K\)-theory torsor via (A). (C) Show that the \(K\)-theory torsor in (B) indeed gives rise via truncation to the graded determinant torsor constructed in [loc. cit.]. S. Saito [“Higher Tate central extensions via \(K\)-theory and infinity-topos theory”, Preprint, arXiv:1405.0923] answered (A) and (B). In this paper under review, the authors address the remaining (C). For this purpose, they use an analogue of a classical result that the index map \([X , Fred(\mathcal{H})] \to K^{\mathrm{top}}(X)\) is isomorphism of groups, where \(Fred(\mathcal{H})\) is the set of Fredholm operators on a separable complex Hilbert space \(\mathcal{H}\) and \(K^{\mathrm{top}}(X)\) is a complex topological \(K\)-theory of a compact Hausdorff space \(X\).

MSC:

19D55 \(K\)-theory and homology; cyclic homology and cohomology
19K56 Index theory

Citations:

Zbl 1108.14012
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References:

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