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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 $\pi_2(Y)$ is trivial. The author turns this into an algebraic question by showing that $\pi_2(Y)$ is the intersection of the terms of the lower central series of the crossed module $\pi_2(Y,Y^1)\to\pi_1(Y^1)$, where $Y^1$ is the 1-skeleton of $Y$. The proof is based on the following algebraic result. Let $\partial'$ and $\partial''$ 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 $\partial'{}^{\text{cr}}$ be the crossed modules induced by $\partial$ and $\partial'$. If the kernel of $\partial^{\text{cr}}$ is trivial, then the kernel of $\partial'{}^{\text{cr}}$ is the intersection of the terms in the lower central series of $\partial'{}^{\text{cr}}$.

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