## 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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