The first stable homotopy groups of motivic spheres
type: Item
instance of: scholarly article
title: The first stable homotopy groups of motivic spheres
zbMATH Open document ID: 1406.14018
author: Oliver Röndigs
author: Markus Spitzweck
author: Paul Arne Østvær
published in: Annals of Mathematics. Second Series
publication date: 2019-01-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/1604.00365
review text: The homotopy groups of the motivic sphere spectrum \(\pi_{p,q}\mathbb{S}\) are fundamental invariants in motivic homotopy theory, which are analogs of the homotopy groups in algebraic topology that in addition carry arithmetic information about fields. A famous result of \textit{F. Morel} [in: Proceedings of the international congress of mathematicians (ICM). Volume II: Invited lectures. Zürich: European Mathematical Society (EMS). 1035--1059 (2006; Zbl 1097.14014)] says that \(\pi_{p,q}\mathbb{S}=0\) when \(p<q\) and \(\pi_{n,n}\mathbb{S}=K^{MW}_n(F)\) is the Milnor-Witt \(K\)-theory, but very few is known on the \(r\)-line \(\pi_{n+r,n}\mathbb{S}\) for \(r>0\). This paper is a solid step towards the understanding of these groups, where the authors are able to compute the \(1\)-line (Theorem 5.5): if \(S\) is a scheme and \(\Lambda\) is a ring such that \((S,\Lambda)\) is a \textit{compatible pair} (Definition 2.1), for example if \(S\) is smooth over a field whose exponential characteristic is invertible in \(\Lambda\), or if \(S\) is regular and \(\Lambda=\mathbb{Q}\), then there is an exact sequence of Nisnevich sheaves over smooth \(S\)-schemes \[ 0\to K^M_{2-n}\otimes\Lambda/24 \to \pi_{n+1,n}\mathbb{S}_\Lambda \to \pi_{n+1,n}f_0(\mathbf{KQ}_\Lambda) \] where \(f_0(\mathbf{KQ}_\Lambda)\) is the effective cover of the hermitian \(K\)-theory spectrum, and the sequence is also right exact for \(n\geq-4\). The main tool is Voevodsky's slice spectral sequence [\textit{V. Voevodsky}, in: Motives, polylogarithms and Hodge theory. Part I: Motives and polylogarithms. Papers from the International Press conference, Irvine, CA, USA, June 1998. Somerville, MA: International Press. 3--34 (2002; Zbl 1047.14012)], which is an analog of the Atiyah-Hirzebruch spectral sequence. Using some convergence results on the slice spectral sequence, the basic idea is to identify the slices of the sphere spectrum (Theorem 2.12) and compute the first differentials in terms of motivic Stee…
reviewed by: Fangzhou Jin
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 55Q45
zbMATH DE Number: 7003144
zbMATH Keywords: stable homotopy of motivic spheres
zbMATH Keywords: slice spectral sequence
zbMATH Keywords: Morel's \(\pi_1\)-conjecture
MaRDI profile type: Publication
OpenAlex ID: W2963102221
arXiv ID: 1604.00365
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The first stable homotopy groups of motivic spheres (DOI)
DOI: 10.4007/ANNALS.2019.189.1.1