Some properties of the Thom spectrum over loop suspension of complex projective space

This note provides a reference for some properties of the Thom spectrum $M\xi$ over $\mathrm{\Omega }\mathrm{\Sigma }\text{CPi}$. Some of this material is used in recent work of Kitchloo and Morava. We determine the $M\xi$-cohomology of $\text{CPi}$ and show that $M{\xi }^{\ast }(\text{CPi})$ injects into power series over the algebra of non-symmetric functions. We show that $M\xi$ gives rise to a commutative formal group law over the non-commutative ring ${\pi }_{\ast }M\xi$. We also discuss how $M\xi$ and some real and quaternionic analogues behave with respect to spectra that are related to these Thom spectra by splittings and by maps.