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 $Mξ$ over $ΩΣ\CPi$. Some of this material is used in recent work of Kitchloo and Morava. We determine the $Mξ$-cohomology of $\CPi$ and show that $Mξ^*(\CPi)$ injects into power series over the algebra of non-symmetric functions. We show that $Mξ$ gives rise to a commutative formal group law over the non-commutative ring $π_*Mξ$. We also discuss how $Mξ$ and some real and quaternionic analogues behave with respect to spectra that are related to these Thom spectra by splittings and by maps.

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