×

Relative derived contravariant functors and the cohomology of groups. (English) Zbl 0697.20042

Author’s abstract: This article investigates the cohomology of a group relative to a collection of normal subgroups. A group-theoretic description of the first relative cohomology is given and used to deduce: a new five term exact sequence; some Hopf type formulas; and a new version of the Universal Coefficient Theorem.
Reviewer: U.Stammbach

MSC:

20J05 Homological methods in group theory
18G25 Relative homological algebra, projective classes (category-theoretic aspects)
18G10 Resolutions; derived functors (category-theoretic aspects)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Barr, M.; Beck, J., Homology and standard constructions, (Lecture Notes in Mathematics, 80 (1969), Springer: Springer Berlin), 245-335
[2] Brown, R.; Ellis, G. J., Hopf formulae for the higher homology of a group, Bull. London Math. Soc., 20, 124-128 (1988) · Zbl 0611.20032
[3] Curtis, E. B., Simplicial homotopy theory, Adv. in Math., 6, 107-204 (1971) · Zbl 0225.55002
[4] Seminar on triples and categorical homology theory, (Lecture Notes in Mathematics, 80 (1969), Springer: Springer Berlin), introduction. · Zbl 1134.18001
[5] Hilton, P. J.; Stammbach, U., A Course in Homological Algebra, (Graduate Texts In Mathematics, 4 (1971), Springer: Springer Berlin) · Zbl 0238.18006
[6] Huebschmann, J., Automorphisms of group extensions and differentials in the Lyndon-Hoschild-Serre spectral sequence, J. Algebra, 72, 296-334 (1981) · Zbl 0443.18018
[7] Keune, F., The relativisation of \(K_2\), J. Algebra, 54, 159-177 (1978) · Zbl 0403.18009
[8] Loday, J.-L., Cohomologie et groupe de Steinberg relatif, J. Algebra, 54, 178-202 (1978) · Zbl 0391.20040
[9] Loday, J.-L., Spaces with finitely many homotopy groups, J. Pure Appl. Algebra, 24, 179-202 (1982) · Zbl 0491.55004
[10] Loday, J.-L., Spaces with finitely many homotopy groups, J. Pure Appl. Algebra, 14, 125-127 (1970)
[11] Steiner, R. J., Resolutions of spaces by cubes of fibrations, J. London Math. Soc., 34, 2, 169-176 (1986) · Zbl 0576.55007
[12] van Osdol, D. H., Long exact sequences in the first variable for algebraic cohomology theories, J. Pure Appl. Algebra, 23, 271-309 (1982) · Zbl 0483.18010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.