On the Power Operations of MU

Abstract

We study the power opearations of the spectrum $MU$, the complex cobordism theory. After reviewing necessary backgrounds, we start by recalling a formula from Justin Noel and Niles Johnson connecting the power operation $P([\mb{CP}^n])$ for $[\mb{CP}^n]\in MU^*$ with the power operation $P_{\mb{CP}}(x)$ for the orientation class $x\in MU^*(\mb{CP}^\infty)$. Next, we find an algorithm calculating the effect of $P$ on a set of polynomial generators $x_i\in MU^*$ from the known formulas coverting $[\mb{CP}^n]$'s to $x_i$'s and the fact that under the canonical projection $q:MU^*\bbra{\alpha}/\bra{p}\rightarrow MU^*\bbra{\alpha}/\ang{p},$ both $q_*P$ and $q_*P_{\mb{CP}}$ become ring homomorphisms. Finally, we display a sample calculation of $P(x_3)$ at $p=2$ with the help of Maple, and provide an application of our calculation where we put some restrictions on possible $E_3$ maps from $MU$ to $BP$(or $BP\langle n\rangle$'s).