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Stable Stems
About this Title
Daniel C. Isaksen, Department of Mathematics, Wayne State University, Detroit, MI 48202, USA
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 262, Number 1269
ISBNs: 978-1-4704-3788-6 (print); 978-1-4704-5511-8 (online)
DOI: https://doi.org/10.1090/memo/1269
Published electronically: December 30, 2019
Keywords: Stable homotopy group,
stable motivic homotopy theory,
May spectral sequence,
Adams spectral sequence,
cohomology of the Steenrod algebra,
Adams-Novikov spectral sequence
MSC: Primary 14F42, 55Q45, 55S10, 55T15; Secondary 16T05, 55P42, 55Q10, 55S30
Table of Contents
Chapters
- 1. Introduction
- 2. The cohomology of the motivic Steenrod algebra
- 3. Differentials in the Adams spectral sequence
- 4. Hidden extensions in the Adams spectral sequence
- 5. The cofiber of $\tau$
- 6. Reverse engineering the Adams-Novikov spectral sequence
- 7. Tables
- 8. Charts
Abstract
We present a detailed analysis of 2-complete stable homotopy groups, both in the classical context and in the motivic context over $\mathbb {C}$. We use the motivic May spectral sequence to compute the cohomology of the motivic Steenrod algebra over $\mathbb {C}$ through the 70-stem. We then use the motivic Adams spectral sequence to obtain motivic stable homotopy groups through the 59-stem. We also describe the complete calculation to the 65-stem, but defer the proofs in this range to forthcoming publications. In addition to finding all Adams differentials, we also resolve all hidden extensions by $2$, $\eta$, and $\nu$ through the 59-stem, except for a few carefully enumerated exceptions that remain unknown. The analogous classical stable homotopy groups are easy consequences.
We also compute the motivic stable homotopy groups of the cofiber of the motivic element $\tau$. This computation is essential for resolving hidden extensions in the Adams spectral sequence. We show that the homotopy groups of the cofiber of $\tau$ are the same as the $E_2$-page of the classical Adams-Novikov spectral sequence. This allows us to compute the classical Adams-Novikov spectral sequence, including differentials and hidden extensions, in a larger range than was previously known.
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