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

Abstract

Using Ravenel's Thom spectrum $X(n)$, we introduce the concept of chromatic defect, which measures how far a spectrum is from being complex-orientable. We compute the chromatic defect of various examples of interest, such as finite spectra, the Real Johnson--Wilson spectra $ER(n)$, fixed points of Morava $E$-theories (with respect to finite subgroups of the Morava stabilizer group), and the connective image of $J$ spectrum. Moreover, an obstruction theory is developed for determining chromatic defect. Having finite chromatic defect is closely related to the existence of analogues of the classical Wood equivalence. We show that such equivalences exist in a wide generality and use them to construct $\mathbb{Z}$-indexed Adams--Novikov towers.