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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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