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 $K{U}_2^{\wedge }$ and other forms of $K$-theory.

13 pages, comments welcome!

References in corpus (1)

• An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology

Cited by in corpus (1)

• Thom spectra, higher $THH$ and tensors in $\infty$-categories