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An algebraic model of fibration with the fiber \(K(\pi,\;n)\)-space. (English) Zbl 0920.55017
Summary: For a fibration with the fiber a \(K(\pi,n)\)-space, the algebraic model as a twisted tensor product of chains of the base with standard chains of \(K(\pi,n)\)-complex is given which preserves multiplicative structure as well. In terms of this model the action of the \(n\)-cohomology of the base with coefficients in \(\pi\) on the homology of the fibration is described.

55R20 Spectral sequences and homology of fiber spaces in algebraic topology
55N10 Singular homology and cohomology theory
55N45 Products and intersections in homology and cohomology
55U99 Applied homological algebra and category theory in algebraic topology
[1] E. H. Brown, Twisted tensor products.Ann. Math. 69 (1959), 223–246. · Zbl 0199.58201
[2] J. P. May, Simplicial objects in algebraic topology.Van Nostrand, Princeton, 1967.
[3] S. MacLane, The homology product inK({\(\Pi\)},n).Proc. Amer. Math. Soc. 5 (1954), 642–651. · Zbl 0059.16405
[4] M. M. Postnikov, Cubical resolutions.Dokl. Akad. Nauk SSSR 118 (1958), 1085–1087. · Zbl 0081.17202
[5] N. Berikashvili, On the obstruction functor.Bull. Georgian Acad. Sci. (to appear). · Zbl 0876.55014
[6] N. Berikashvili, An algebraic model of the Postnikov construction,Bull. Georgian Acad. Sci. 152 (1995). · Zbl 0877.55015
[7] D. M. Kan, Abstract homotopy.Proc. Natl. Acad. Sci. USA 41 (1955), 1092–1096. · Zbl 0065.38601
[8] J. Milnor, The geometric realization of a semi-simplicial complex.Ann. Math. 65 (1957), 357–362. · Zbl 0078.36602
[9] J. B. Giever, On the equivalence of two singular homology theories.Ann. Math. 51 (1950), 178–191. · Zbl 0035.38801
[10] S. T. Hu, On the realizability of homotopy groups and their operations.Pacific J. Math. 1 (1951), 583–602. · Zbl 0044.19902
[11] J. P. Serre, Homologie singuliere des éspaces fibrés, applications.Ann. Math. 54 (1951), 429–505. · Zbl 0045.26003
[12] N. Berikashvili, On the homology theory of fibrations.Bull. Acad. Sci. Georgia 139 (1990), 17–19. · Zbl 0723.55008
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