On the differentials in the spectral sequence of a~group extension

Yu. V. Kuz'min

Sbornik. Mathematics, Tome 61 (1988) no. 1, pp. 49-63

Résumé

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.