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Van Kampen theorems for diagrams of spaces. (English) Zbl 0622.55009

Consider a commutative diagram of spaces formed by the vertices and edges of an \(n\)-cube. The fundamental groups of its iterated homotopy fibres form an algebraic object called a cat\(^ n\)-group. The authors show that under suitable connectedness hypotheses the functor from cubical diagrams to cat\(^ n\)-groups takes homotopy colimits to colimits. They study some related algebraic constructions, in particular a tensor product of non-abelian groups. There are various applications. Under suitable conditions a triad homotopy group \(\pi_ 3(X;A,B)\) (where \(X=A\cup B)\) is isomorphic to \(\pi_ 2(X,A)\otimes \pi_ 2(X,B)\), even if the groups involved are not abelian. The third homotopy group of a suspension is computed and exact sequences involving the homology of group extensions are constructed.
Reviewer: R.Steiner

MSC:

55Q05 Homotopy groups, general; sets of homotopy classes
57M05 Fundamental group, presentations, free differential calculus
55Q15 Whitehead products and generalizations
55P20 Eilenberg-Mac Lane spaces
20J05 Homological methods in group theory
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