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Whitehead’s question and precrossed modules. (Question de Whitehead et modules précroisés.) (French) Zbl 0865.57005

Let X be an aspherical 2-dimensional CW complex with a single 0-cell, and let Y be a subcomplex. Whitehead’s question asks whether Y is also aspherical, or, equivalently, whether the homotopy group π 2 (Y) is trivial. The author turns this into an algebraic question by showing that π 2 (Y) is the intersection of the terms of the lower central series of the crossed module π 2 (Y,Y 1 )π 1 (Y 1 ), where Y 1 is the 1-skeleton of Y.

The proof is based on the following algebraic result. Let ' and '' be totally free pre-crossed modules over the same group P, and let be their coproduct (as pre-crossed P-modules). Let cr and ' cr be the crossed modules induced by and ' . If the kernel of cr is trivial, then the kernel of ' cr is the intersection of the terms in the lower central series of ' cr .


MSC:
57M20Two-dimensional complexes (manifolds)
18G30Simplicial sets; simplicial objects in a category
20F38Other groups related to topology or analysis