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

Abstract

This note provides a reference for some properties of the Thom spectrum $M\xi$ over $\Omega\Sigma\CPi$. Some of this material is used in recent work of Kitchloo and Morava. We determine the $M\xi$-cohomology of $\CPi$ and show that $M\xi^*(\CPi)$ 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 $\pi_*M\xi$. 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.