Exotic Multiplications on Periodic Complex Bordism

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.