The order of the differentials in the Atiyah-Hirzebruch spectral sequence
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.
bounds on the orders of the elements in the images of the differentials
scientific article; zbMATH DE number 149777
scientific article