*(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 ${\pi}_{2}(Y,{Y}^{1})\to {\pi}_{1}\left({Y}^{1}\right)$, 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 ${\partial}^{\text{'}}{}^{\text{cr}}$.