On the Power Operations of MU

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{α}/\bra{p}\rightarrow MU^*\bbra{α}/\ang{p}, both q_*P and q_*P_\mb{CP} become ring homomorphisms. Finally, we display a sample calculation of P(x₃) at $p=2$ with the help of Maple, and provide an application of our calculation where we put some restrictions on possible E₃ maps from $MU$ to $BP$(or $BP\langle n\rangle$'s).

Author dc:creator

• Gu, Zeshen

Language dc:language.iso

en

Handle dc:identifier.uri

https://hdl.handle.net/11299/241394

OAI identifier oai:identifier

oai:conservancy.umn.edu:11299/241394