Chromatic defect, Wood's theorem, and higher real K-theories

Let X(n) be Ravenel's Thom spectrum over Ω SU(n). We say a spectrum E has chromatic defect n if n is the smallest positive integer such that E⊗ X(n) is complex orientable. We compute the chromatic defect of various examples of interest: finite spectra, the Real Johnson--Wilson theories ER(n), the fixed points EOₙ(G) of Morava E-theories with respect to a finite subgroup G of the Morava stabilizer group, and the connective image of J spectrum j. Having finite chromatic defect is closely related to the existence of analogues of the classical Wood splitting ko⊗ C(η) ku. We show that such splittings exist in quite a wide generality for fp spectra E. When E participates in such a splitting, E admits a Z-indexed Adams--Novikov tower, which may be used to deduce differentials in the Adams--Novikov spectral sequence of E.