# Exotic Multiplications on Periodic Complex Bordism

## TL;DR

This paper compares two constructions of periodic complex bordism, showing they are different as $bE_$-rings but share the same underlying structure, and both orient certain $K$-theories.

## Contribution

It demonstrates the difference between Snaith's and the Thom spectrum constructions of periodic complex bordism as $bE_$-rings, and their common orientation properties.

## Key findings

- The two $bE_$-ring structures are not equivalent.
- Both structures $bE_$-orient $KU_2^{}$ and related $K$-theories.
- Underlying $bE_2$-rings of the two constructions are equivalent.

## Abstract

Victor Snaith gave a construction of periodic complex bordism by inverting the Bott element in the suspension spectrum of $BU$. This presents an $\mathbb{E}_\infty$ structure on periodic complex bordism by different means than the usual Thom spectrum definition of the $\mathbb{E}_\infty$-ring $MUP$. Here, we prove that these two $\mathbb{E}_\infty$-rings are in fact different, though the underlying $\mathbb{E}_2$-rings are equivalent. Nonetheless, we prove that both rings $\mathbb{E}_\infty$-orient $KU_2^{\wedge}$ and other forms of $K$-theory.