Multiplicative Structure in the Stable Splitting of $\Omega S{L}_n(\mathbb{C})$

Abstract

The space of based loops in $S{L}_n(\mathbb{C})$, also known as the affine Grassmannian of $S{L}_n(\mathbb{C})$, admits an ${\mathbb{E}}_2$ or fusion product. Work of Mitchell and Richter proves that this based loop space stably splits as an infinite wedge sum. We prove that the Mitchell--Richter splitting is coherently multiplicative, but not ${\mathbb{E}}_2$. Nonetheless, we show that the splitting becomes ${\mathbb{E}}_2$ after base-change to complex cobordism. Our proof of the ${\mathbb{A}}_{\infty }$ splitting involves on the one hand an analysis of the multiplicative properties of Weiss calculus, and on the other a use of Beilinson--Drinfeld Grassmannians to verify a conjecture of Mahowald and Richter. Other results are obtained by explicit, obstruction-theoretic computations.

32 pages. Comments welcome!

References in corpus (3)

• Higher Topos Theory
• An introduction to affine Grassmannians and the geometric Satake equivalence
• Stable $\infty$-Operads and the multiplicative Yoneda lemma

Cited by in corpus (2)

• Exotic Multiplications on Periodic Complex Bordism
• A model for the E3 fusion-convolution product of constructible sheaves on the affine Grassmannian