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

TL;DR

This paper investigates the properties of the Thom spectrum over the loop suspension of complex projective space, including its cohomology, algebraic structures, and relations to other spectra, providing foundational insights for related topological studies.

Contribution

It determines the $M\xi$-cohomology of $\CPi$, shows its injection into power series over non-symmetric functions, and explores its formal group law over a non-commutative ring.

Findings

$M\xi$-cohomology of $\CPi$ is explicitly computed.

$M\xi^*(\CPi)$ injects into power series over non-symmetric functions.

$M\xi$ induces a formal group law over a non-commutative ring.

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.