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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 ${\pi }_{2}\left(Y\right)$ is trivial. The author turns this into an algebraic question by showing that ${\pi }_{2}\left(Y\right)$ is the intersection of the terms of the lower central series of the crossed module ${\pi }_{2}\left(Y,{Y}^{1}\right)\to {\pi }_{1}\left({Y}^{1}\right)$, where ${Y}^{1}$ is the 1-skeleton of $Y$.

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

##### MSC:
 57M20 Two-dimensional complexes (manifolds) 18G30 Simplicial sets; simplicial objects in a category 20F38 Other groups related to topology or analysis