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Zbl 0865.57005
Conduché, Daniel
Whitehead's question and precrossed modules. (Question de Whitehead et modules précroisés.) (French)
[J] Bull. Soc. Math. Fr. 124, No. 3, 401-423 (1996). ISSN 0037-9484

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 $Y$.\par 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 $\partial'{}^{\text{cr}}$.
[R.J.Steiner (Glasgow)]

MSC 2000:: *57M20 Two-dimensional complexes
18G30 Simplicial objects in a category
20F38 Other groups related to topology or analysis

Keywords: CW complex; aspherical; homotopy group; lower central series; crossed module

Cited in: Zbl 1089.57004

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