The order of the differentials in the Atiyah-Hirzebruch spectral sequence. (English) Zbl 0768.55012
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 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 H. Cartan, J. C. Moore, R. Thom and 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.
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 H. Cartan, J. C. Moore, R. Thom and 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.
Reviewer: D.W.Kahn (Minneapolis)
MSC:
| 55T25 | Generalized cohomology and spectral sequences in algebraic topology |
| 55N20 | Generalized (extraordinary) homology and cohomology theories in algebraic topology |
Keywords:
Atiyah-Hirzebruch spectral sequence; bounds on the orders of the elements in the images of the differentials; collapsed spectral sequenceReferences:
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