Stable homotopy groups of spheres. A computer-assisted approach
type: Item
instance of: scholarly article
title: Stable homotopy groups of spheres. A computer-assisted approach
zbMATH Open document ID: 0688.55010
author: Stanley O. Kochman
published in: Lecture Notes in Mathematics
publication date: 1992-09-17T00:00:00Z
review text: From the author's preface: ``This work develops the theoretical basis for an efficient method for the inductive calculation of the stable homotopy groups of spheres, \(\pi^ S_*\). Most of the steps of this method are algorithmic and are done by computer. We will apply this method to compute the first 64 stable stems. This method is based upon the analysis of the Atiyah-Hirzebruch spectral sequence: \[ (*)\quad E^ 2_{n,t}=H_ nBP\otimes \pi^ S_ t\Rightarrow \pi_{n+t}BP. \] \(H_*BP\) and \(\pi_*BP\) are well known. Moreover, the Hurewicz homomorphism h: \(\pi_*BP\to H_*BP\) is a monomorphism. Therefore, \(E^{\infty}_{n,t}=0\) if \(t\neq 0\), and \(E^{\infty}_{n,0}=h(\pi_ nBP)\) which is also well known. If \(\pi^ S_ t\) is known for \(t<T\) then, with the exception of one step, it is algorithmic to deduce the composition series Image \([d^ r: E^ r_{r,T-r+1}\to E^ r_{0,T}]\), \(2\leq r\leq T+1\), of \(\pi^ S_ T\). The determination of \(\pi^ S_ T\) from the composition series, the solution of the ``additive extension problem'', is accomplished using Toda brackets. A distinctive feature of this method is that all the hard computations are done by computer. This includes the determination of differentials using Quillen operations and the computation of \[ E^{r+1}_{N,t}=Kernel\quad [d^ r: E^ r_{N,t}\to E^ r_{N- r,t+r-1}]/Image\quad [d^ r: E^ r_{N+r,t-r+1}\to E^ r_{N,t}]. \] On the other hand there are two key steps which require human intervention in the computation of each \(\pi^ S_ T:\) (1) the matching of the list of ``new'' elements in degree \(T+1\) which are hit by differentials with the list of ``new'' elements in degree \(T+2\) on which nonzero differentials orginate; (2) the solution of the additive extension problems. Chapter 1 is devoted to the exposition of the background of this computation and to a detailed description of the method we will use. Even the most experienced reader should read the exposition of our notation for elements of the stable stems at the…
reviewed by: Mamoru Mimura
Mathematics Subject Classification ID: 55Q45
Mathematics Subject Classification ID: 55-02
Mathematics Subject Classification ID: 55-04
zbMATH DE Number: 45857
zbMATH Keywords: Atiyah-Hirzebruch spectral sequence
zbMATH Keywords: Hurewicz homomorphism
zbMATH Keywords: Toda brackets
zbMATH Keywords: Quillen operations
zbMATH Keywords: Mahowald elements
zbMATH Keywords: mod 2 Adams spectral sequence
Wikidata QID: Q56456314
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0083795
OpenAlex ID: W2482288358
DOI: 10.1007/BFB0083795
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description: scientific article; zbMATH DE number 45857