# The circle action on topological Hochschild homology of complex cobordism and the Brown-Peterson spectrum

## TL;DR

This paper explicitly describes the generators and the action of the circle operator on the topological Hochschild homology of complex cobordism and Brown-Peterson spectra, deepening understanding of their algebraic structures.

## Contribution

It provides explicit exterior generators and computes the circle action on THH of MU and BP spectra, clarifying their algebraic and homotopical properties.

## Key findings

- Explicit exterior generators for π_* THH(MU) and π_* THH(BP)
- Calculation of the σ-operator action on these generators
- Expression of actions on π_* MU and π_* BP via right units in Hopf algebroids

## Abstract

We specify exterior generators for $\pi_* THH(MU) = \pi_*(MU) \otimes E(\lambda'_n \mid n\ge1)$ and $\pi_* THH(BP) = \pi_*(BP) \otimes E(\lambda_n \mid n\ge1)$, and calculate the action of the $\sigma$-operator on these graded rings. In particular, $\sigma(\lambda'_n) = 0$ and $\sigma(\lambda_n) = 0$, while the actions on $\pi_*(MU)$ and $\pi_*(BP)$ are expressed in terms of the right units $\eta_R$ in the Hopf algebroids $(\pi_*(MU), \pi_*(MU \wedge MU))$ and $(\pi_*(BP), \pi_*(BP \wedge BP))$, respectively.