The order of the differentials in the Atiyah-Hirzebruch spectral sequence
type: Item
instance of: scholarly article
title: The order of the differentials in the Atiyah-Hirzebruch spectral sequence
zbMATH Open document ID: 0768.55012
DOI: 10.1007/BF00966117
author: Dominique Arlettaz
published in: \(K\)-Theory
publication date: 1993-04-01T00:00:00Z
review text: The Atiyah-Hirzebruch spectral sequence of a CW-complex \(X\), with coefficients in a spectrum \(F\), is a spectral sequence which begins with \(H_ s(X,\pi_ tF)\) and converges to \(F_{s+t}(X)\). Because the rational, stable Postnikov terms of a nice space are products of Eilenberg-MacLane spaces, the rational spectral sequence can be seen to collapse. This was first observed by \textit{A. Dold} [Halbexakte Homotopiefunktoren, Lect. Notes Math. 12 (1966; Zbl 0136.00801)] in the contravariant case. The present author gives universal bounds on the orders of the elements in the images of the differentials in the Atiyah-Hirzebruch sequence. These results depend, in a fundamental way, on the work of \textit{H. Cartan}, \textit{J. C. Moore}, \textit{R. Thom} and \textit{J. P. Serre} [Algèbres d'Eilenberg-MacLane et homotopie, École Norm. Supér., Seminaire Henri Cartan 1954/1955 (1955; Zbl 0067.15601)]. This work shows that if \(\pi\) is a nice Abelian group, \(p\) is a prime, and \(n<i<2n-1\), then \(H_ i(\pi,n;Z)\) contains no element of order \(p^ 2\). In addition, in certain cases the author is also able to show that the extension problem, arising from a collapsed spectral sequence, is trivial.
reviewed by: Donald W. Kahn
Mathematics Subject Classification ID: 55T25
Mathematics Subject Classification ID: 55N20
zbMATH DE Number: 149777
zbMATH Keywords: Atiyah-Hirzebruch spectral sequence
zbMATH Keywords: bounds on the orders of the elements in the images of the differentials
zbMATH Keywords: collapsed spectral sequence
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00966117
OpenAlex ID: W2055998277
description: scientific article; zbMATH DE number 149777