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Cohomology with coefficients in symmetric cat-groups. An extension of Eilenberg–MacLane's classification theorem

Published online by Cambridge University Press:  24 October 2008

M. Bullejos ,
P. Carrasco  and
A. M. Cegarra
M. Bullejos
Affiliation:
Departamento de Algebra, Universidad de Granada, 18071 Granada, Spain
P. Carrasco
Affiliation:
Departamento de Algebra, Universidad de Granada, 18071 Granada, Spain
A. M. Cegarra
Affiliation:
Departamento de Algebra, Universidad de Granada, 18071 Granada, Spain
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Abstract

In this paper we use Takeuchy–Ulbrich's cohomology of complexes of categories with abelian group structure to introduce a cohomology theory for simplicial sets, or topological spaces, with coefficients in symmetric cat-groups . This cohomology is the usual one when abelian groups are taken as coefficients, and the main topological significance of this cohomology is the fact that it is equivalent to the reduced cohomology theory defined by a Ω-spectrum, {}, canonically associated to . We use the spaces to prove that symmetric cat-groups model all homotopy type of spaces X with Πi(X) = 0 for all i ╪ n, n + 1 and n ≥ 3, and then we extend Eilenberg–MacLane's classification theorem to those spaces: .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 163 - 189
Copyright
Copyright © Cambridge Philosophical Society 1993

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