A nontrivial product of filtration \(s+ 5\) in the stable homotopy of spheres
type: Item
instance of: scholarly article
title: A nontrivial product of filtration \(s+ 5\) in the stable homotopy of spheres
zbMATH Open document ID: 1120.55012
publication date: 2007-06-07T00:00:00Z
review text: Let \(\beta_{s+2}\) and \(\gamma_{p^{n-2}/p^{n-2}-1}\) be the beta and the gamma elements of the stable homotopy groups of spheres at the prime \(p\) greater than three constructed by \textit{L. Smith} [Am. J. Math. 99, 418-436 (1977; Zbl 0389.55017)] and \textit{J. Lin} [Hiroshima Math. J. 31, No.3, 477-492 (2001; Zbl 0997.55018)], respectively. In this paper, the author essentially shows the non-triviality of the product of these elements by observing the \(E_2\)-term of the Adams spectral sequence. The main tool he uses to determine the \(E_2\)-term is the May spectral sequence. First he shows that the element \(a_2^sh_{2,0}h_{1,1}\) of the \(E_2\)-term detects the beta element, and then the non-triviality of the product \(a_2^sh_{2,0}h_{1}(b_0h_n +h_1b_{n-1})\) in the \(E_2\)-term. Here \(a_2^s\) denotes the \(s\)-fold Yoneda product of \(a_2\), and the element \(b_0h_n +h_1b_{n-1}\) in the \(E_2\)-term detects the gamma element, \textit{J. Lin} [op. cit.]. The main result, which says that \(\beta_{s+2}\gamma_{p^{n-2}/p^{n-2}-1}\) is essential in the stable homotopy of spheres for \(n\geq 3\), \(0\leq s< p-3\), follows by showing that the original \(E_2\)-terms of the Adams differentials that may hit the product are all trivial.
reviewed by: Katsumi Shimomura
Mathematics Subject Classification ID: 55Q45
Mathematics Subject Classification ID: 55T15
zbMATH DE Number: 5162316
zbMATH Keywords: stable homotopy of spheres
zbMATH Keywords: Adams spectral sequence
zbMATH Keywords: May spectral sequence
author: Liu Xiugui
MaRDI profile type: Publication
full work available at URL: s10114-005-0548-y
OpenAlex ID: W2145372163
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DOI: 10.1007/S10114-005-0548-Y
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5162316