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.

中文翻译：

---

周期复杂bordism的奇异乘法

Victor Snaith通过反转B悬置谱中的Bott元素，给出了周期复杂bordism的构造。 $乙ü$。这代表了${\mathbb{Ë}}_{\infty }$ 周期性复杂bordism的结构不同于通常的Thom频谱定义 ${\mathbb{Ë}}_{\infty }$-环 $\mathrm{中号}üP$。在这里，我们证明这两个${\mathbb{Ë}}_{\infty }$环实际上是不同的，尽管底层 ${\mathbb{Ë}}_2$环是等效的。尽管如此，我们证明两个环${\mathbb{Ë}}_{\infty }$-东方 $ķ{ü}_2^{\wedge }$ 和其他形式 $ķ$-理论。