Jeremy Hahn, Allen Yuan

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 $K{U}_{2}^{\wedge}$ and other forms of $K$‐theory.