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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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##### References:
  DOI: 10.1090/S0273-0979-1994-00438-0 · Zbl 0857.55003 · doi:10.1090/S0273-0979-1994-00438-0  Residues and duality (1966)  Mem. Amer. Math. Soc. 113 pp 178– (1995)  DOI: 10.4007/annals.2005.162.777 · Zbl 1108.55009 · doi:10.4007/annals.2005.162.777  DOI: 10.1007/s00220-006-0181-3 · Zbl 1126.81045 · doi:10.1007/s00220-006-0181-3  Theory Appl. Categ. 11 pp 107– (2003)  Equivariant homotopy and cohomology theory 91 (1996)  DOI: 10.2977/prims/1201011783 · Zbl 1136.55009 · doi:10.2977/prims/1201011783  DOI: 10.1023/A:1012785711074 · Zbl 1005.19003 · doi:10.1023/A:1012785711074  J. Differential Geom. 70 pp 329– (2005)  DOI: 10.1112/S0024610799007784 · Zbl 0947.55013 · doi:10.1112/S0024610799007784  Topology and representation theory (Evanston, IL, 1992), Contemp. Math. 158 pp 89– (1994)  DOI: 10.1090/S0002-9947-07-04204-3 · Zbl 1125.55002 · doi:10.1090/S0002-9947-07-04204-3  Trans. Amer. Math. Soc. 347 pp 1841– (1995)  Mathematical Surveys and Monographs 47 (1997)  K -Theory 35 pp 213– (2005)  Inst. Hautes Études Sci. Publ. Math. pp 5– (1971)  DOI: 10.1090/S0002-9947-06-04201-2 · Zbl 1111.55009 · doi:10.1090/S0002-9947-06-04201-2  Osaka J. Math. 12 pp 305– (1975)  DOI: 10.1016/0040-9383(79)90018-1 · Zbl 0417.55007 · doi:10.1016/0040-9383(79)90018-1  DOI: 10.1016/S0040-9383(99)00049-X · Zbl 0957.55003 · doi:10.1016/S0040-9383(99)00049-X  Doc. Math. 17 pp 271– (2012)  DOI: 10.1093/qmath/17.1.367 · Zbl 0146.19101 · doi:10.1093/qmath/17.1.367  Mem. Amer. Math. Soc. 192 pp 137– (2008)  Universal coefficient theorems for K-theory, mimeographed notes (1969)  DOI: 10.1090/S0894-0347-96-00174-9 · Zbl 0864.14008 · doi:10.1090/S0894-0347-96-00174-9  DOI: 10.1007/BF02621873 · doi:10.1007/BF02621873
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