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\(K\)-theory, reality, and duality. (English) Zbl 1325.55003
Let \(I_{{\mathbb Q}/{\mathbb Z}}\) and \(I_{\mathbb Q}\) denote the spectra representing cohomology theories \(X \mapsto \text{Hom}(\pi_{-*}X, {\mathbb Q}/{\mathbb Z})\) and \(X \mapsto \text{Hom}(\pi_{-*}X, {\mathbb Q})\), respectively (wherein the latter is the rational Eilenberg-MacLane spectrum) and let \(I_{\mathbb Z}\) denote the homotopy fiber of the natural map \(I_{\mathbb Q} \to I_{{\mathbb Q}/{\mathbb Z}}\). Let \(I_{\mathbb Z}X\) be the function spectrum \(F(X, I_{\mathbb Z})\), which is called the Anderson dual of \(X\). Then the main theorem of this paper can be stated as follows: The Anderson dual \(I_{\mathbb Z}KU\) is \(C_2\)-equivariantly equivalent to \(\Sigma^4KU\) where \(C_2\) acts on \(KU\) by complex conjugation. For the proof of this theorem the authors begin by calculating the homotopy groups of \(I_{\mathbf Z}KU\) and obtain that they are \({\mathbb Z}\) in even degrees and 0 in odd degrees. If we let \(d : S^4 \to I_{\mathbb Z}KU\) denote the spectrum map representing a generator of \(\pi_4I_{\mathbb Z}KU\), then it can be extended to a map \(\psi : S^4\wedge KU \to I_{\mathbb Z}KU\) using the \(KU\)-module structure on \(I_{\mathbb Z}KU\). We see that as a matter of fact this map gives an equivalence \(S^4\wedge KU \simeq I_{\mathbb Z}KU\). The theorem above can be proved by showing that this map \(\psi\), or equivalently \(d\) has an equivariant refinement. In order to prove this fact we are required to establish an equivalence such that \(I_{\mathbb Z}KO\simeq\Sigma^4KO\). Incidentally, as a consequence of the use of the results obtained in proving the theorem, the authors find that there exists a dualizing \(KO\)-module relative to the unit map \(u : S \to KO\). That is, the forgetful functor \(u_*\) from \(KO\)-modules to spectra has a right adjoint \(u^!=F(KO, -)\) such that \(F(KO, A)\simeq F(I_{\mathbb Z}A, \Sigma^4KO)\) for a spectrum \(A\). In addition, there is presented an application to the exotic \(K(1)\)-local Picard group \(\kappa_1\), in which it is proved that \(\kappa_1\cong {\mathbb Z}/2\).

MSC:
55N15 Topological \(K\)-theory
19L99 Topological \(K\)-theory
55M05 Duality in algebraic topology
55U30 Duality in applied homological algebra and category theory (aspects of algebraic topology)
55P91 Equivariant homotopy theory in algebraic topology
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