Rational algebraic \(K\)-theory of topological \(K\)-theory. (English) Zbl 1260.19004
In this paper the authors study the algebraic \(K\)-theory \(K(ku)\) of the connective complex \(K\)-theory spectrum \(ku\), with the aim of computing it rationally. There are maps
\[
BBU_\otimes \to BGL_1(ku) \to BGL_\infty(ku) \to BGL_\infty(ku)^+
\]
of spectra induced by units where \(BBU_\otimes\) denotes the first delooping of the space \(BU_\otimes\) representing virtual compex line bundles. Write \(w : BBU_\otimes \to K(ku)\) for the composition of these maps and the inclusion
\[
BGL_\infty(ku)^+\cong \{ 1 \}\times BGL_\infty(ku)^+ \to \mathbb Z\times BGL_\infty(ku)^+\simeq K(ku).
\]
Let \(\pi : K(ku) \to K(\mathbb Z)\) be the map induced in \(K\)-theory by the zeroth Postnikov section \(ku \to H\mathbb Z\). The authors prove that after rationalization, the sequence
\[
BBU_\otimes \overset{w}{\longrightarrow} K(ku) \overset{\pi}{\longrightarrow} K(\mathbb Z)
\]
is a split homotopy fibre-sequence where the splitting of \(\pi\) is given by the rationalization of \(K(S) \to K(ku)\) induced by the unit \(S \to ku\). Also they show that the usual matrix determinant induces a rational determinant map \(\det_\mathbb Q:BGL_\infty(ku) \to (BBU_\otimes)_\mathbb Q\), which provides a splitting of \(w\). In connection with this map it is noted that in the integral case such a determinant map does not exist.
Let \(\mathcal{E} : X \to K(ku)\) be a map from a base space \(X\), which may be thought of as a virtual two-vector bundle over \(X\) introduced by N. A. Baas et al. in [Lond. Math. Soc. Lect. Note Ser. 308, 18–45 (2004; Zbl 1106.55004)]. Using the splitting of \(K(ku)_\mathbb Q\) above, the authors provide a geometric interpretation of this bundle. It is stated as follows. We call the composite map \(|\mathcal{E}|=\det_\mathbb Q\circ \mathcal{E} : X \to (BBU_\otimes)_ Q\) the rational determinant bundle of \(\mathcal{E}\) and call a virtual line bundle \(\mathcal{H} : \mathcal{L}X \to BU_\otimes\) over the free loop space \(\mathcal{L}X=\mathrm{Map}(S^1, X)\) the “anomaly bundle”. Then to specify the rational determinant bundle is equivalent to specifying the rational anomaly bundle.
This paper also contains a computation of the rational algebraic \(K\)-theory \(K(KU)\) of the periodic complex \(K\)-theory spectrum \(KU\).
Let \(\mathcal{E} : X \to K(ku)\) be a map from a base space \(X\), which may be thought of as a virtual two-vector bundle over \(X\) introduced by N. A. Baas et al. in [Lond. Math. Soc. Lect. Note Ser. 308, 18–45 (2004; Zbl 1106.55004)]. Using the splitting of \(K(ku)_\mathbb Q\) above, the authors provide a geometric interpretation of this bundle. It is stated as follows. We call the composite map \(|\mathcal{E}|=\det_\mathbb Q\circ \mathcal{E} : X \to (BBU_\otimes)_ Q\) the rational determinant bundle of \(\mathcal{E}\) and call a virtual line bundle \(\mathcal{H} : \mathcal{L}X \to BU_\otimes\) over the free loop space \(\mathcal{L}X=\mathrm{Map}(S^1, X)\) the “anomaly bundle”. Then to specify the rational determinant bundle is equivalent to specifying the rational anomaly bundle.
This paper also contains a computation of the rational algebraic \(K\)-theory \(K(KU)\) of the periodic complex \(K\)-theory spectrum \(KU\).
Reviewer: Haruo Minami (Nara)
MSC:
| 19L99 | Topological \(K\)-theory |
| 55N15 | Topological \(K\)-theory |
| 18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |
| 19D99 | Higher algebraic \(K\)-theory |
| 19D55 | \(K\)-theory and homology; cyclic homology and cohomology |
Keywords:
algebraic \(K\)-theory; rational homotopy; topological \(K\)-theory; determinants; 2-vector bundles; bordism spectraCitations:
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