Stable homotopy groups of spheres. A computer-assisted approach
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 end of that chapter. In Chapter 2 we develop the three and four-fold Toda bracket methods which are used to solve extension problems. In Chapter 3 we give a global computation of the spectral sequence in the first eight rows. In higher rows our computations are inductive and rarely achieve a global understanding of the rows beyond the range of our computations. In Chapter 4 we recall some facts about the Image of J and use them to compute all the differentials which originate on \(E_ r^{n,0}\) for \(n\leq 70\). Chapters 5 to 7 contain our calculations of the first 64 stable stems. In Chapter 8 we identify the elements \(\theta_ 4\in \pi^ S_{30}\) and \(\theta_ 5\in \pi^ S_{62}\) of Arf invariant one as well as the Mahowald elements \(\eta_ 5\in \pi^ S_{32}\) and \(\eta_ 6\in \pi^ S_{64}\). The new proof that \(\theta_ 5\) exists and has order two is based upon Mahowald's ideas and the computations of this paper. It is a rewording of a detailed proof which Mahowald sent to me. We also show that \(\eta_ 6\) has order four. We conclude with Appendices 1-4, 7 which contain tables that summarize and give references for all the computations of this paper. In the fifth appendix, we discuss the Fortran computer programs which are used in this computation. A copy of the program listings is available from the author. The most important output of these programs is contained in the last sections of Chapters 4-7. The sixth appendix depicts the mod 2 Adams spectral sequence through degree 64.''
scientific article; zbMATH DE number 45857