Abstract
The space of based loops in SL n (C), also known as the affine Grassmannian of SL n (C), admits an 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 E 2 . Nonetheless, we show that the splitting becomes E 2 after base-change to complex cobordism. Our proof of the A ∞ 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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 412-455 |
| Number of pages | 44 |
| Journal | Advances in Mathematics |
| Volume | 348 |
| DOIs | |
| State | Published - May 25 2019 |
Funding
Acknowledgments. The authors thank Greg Arone, Tom Bachmann, Lukas Brantner, Dennis Gaitsgory, Akhil Mathew, Haynes Miller, Denis Nardin, and Bill Richter for helpful conversations and the anonymous referee for numerous clarifying comments. We were saddened to learn of the passing of Steve Mitchell during the preparation of this manuscript–his papers were a deep inspiration. Special thanks are due to the authors’ PhD advisors, Mike Hopkins and Jacob Lurie, both for their mathematical expertise and their consistent encouragement; many of the ideas in this paper grew out of their suggestions. Special thanks are also due to Justin Campbell, James Tao, David Yang, and Yifei Zhao, all of whom spent numerous and invaluable hours answering naive questions about the Beilinson–Drinfeld Grassmannian. The authors were supported by NSF Graduate Fellowships under Grants DGE-1144152 and 1122374.
Keywords
- Affine Grassmannian
- Homotopy theory
- Stable splitting
- Structured ring spectrum
ASJC Scopus subject areas
- General Mathematics