Spectral complications in cohomology computations. (English) Zbl 0553.20031

Contributions to group theory, Contemp. Math. 33, 11-23 (1984).
[For the entire collection see Zbl 0539.00007.]
R. C. Lyndon in his 1946 Harvard Ph. D. Thesis [cf. also Duke Math. J. 15, 271-292 (1948; Zbl 0031.19802)] computed the cohomology groups of a group extension in terms of the cohomology groups of the factors of that extension. The notion of a spectral sequence was partially anticipated in Lyndon’s work. The author in the present paper very beautifully comments on that anticipation of spectral sequences in Lyndon’s work and also brings out as to why spectral sequences must arise in the situation of extensions and in the corresponding situations for fibre spaces. The author in the last two sections mentions the contribution of other researches on the spectral sequences and points out (as follows from L. Evens [Trans. Am. Math. Soc. 212, 269-277 (1975; Zbl 0331.18022)], F. R. Beyl [Bull. Sci. Math., II. Sér. 105, 417-434 (1981; Zbl 0465.18009)] and an unpublished result of D. W. Barnes) that several different filtrations used by G. Hochschild and J.-P. Serre [Trans. Am. Math. Soc. 74, 110-134 (1953; Zbl 0050.021)] give the same well known Lyndon-Hochschild-Serre spectral sequence.
Reviewer: L.R.Vermani


20J05 Homological methods in group theory
20E22 Extensions, wreath products, and other compositions of groups
18G40 Spectral sequences, hypercohomology
55R20 Spectral sequences and homology of fiber spaces in algebraic topology