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On the differentials in the spectral sequence of a group extension

Yu. V. Kuz'min


Abstract: Let $1\to A\to G\to B\to1$ be a group extension in which $A$ is a torsion-free Abelian group. The concept of the $q$th-order characteristic class is introduced. This is an exact sequence of length 2 defined explicitly in terms of the original extension, and it coincides with the usual characteristic class when $q=0$.
The main result is that the differentials $d^2_{pq}$ in the spectral sequence of the extension converging to the homology $H_*(G,Z)$ coincide with multiplication by the $q$th-order characteristic class. Analogous results can be formulated also for cohomology.
Bibliography: 11 titles.

UDC: 512

MSC: Primary 20E22, 20J05, 18G40; Secondary 20E06, 18G10

Received: 03.01.1986

 English version: Mathematics of the USSR-Sbornik, 1988, 61:1, 49–63

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