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The object of this paper is to calculate the Morava K-theory of BU〈6〉, the 5-connected cover of BU. Together with M. J. Hopkins, the authors have calculated the E-theory of BU〈6 〉 for more general complex oriented E, but that result requires a more computational proof and also is only available in preprint form. Note that the word “calculate ” is used in an algebro-geometric sense; rather than computing the Hopf algebra E0BU〈6〉, the authors compute the scheme represented by it. The result says that, for E a 2-periodic Morava K-theory spectrum, spec (E0BU〈6〉) is the scheme of rigid cubical structures on the trivial Gm torsor over the formal group of E, where Gm denotes the multiplicative group. Obviously this calculation is not very explicit, but, on the other hand, it gives very good qualitative information that has been used by the authors and Hopkins to shed considerable light on elliptic cohomology. To give an idea of the proof, note that the Atiyah-Hirzebruch spectral sequence shows that the fibration K(Z, 3) → BU〈6 〉 → BSU gives rise to a short exact sequence of abelian Hopf algebras in E-homology. The authors construct an analogous sequence involving Weil pairings, cubical structures, and symmetric 2-cocycles, and a map from the scheme version of the first exact sequence to the second. Reviewed by Mark Hove

Year: 2013

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