Exotic Multiplications on Periodic Complex Bordism

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