Let be an aspherical 2-dimensional CW complex with a single 0-cell, and let be a subcomplex. Whitehead’s question asks whether is also aspherical, or, equivalently, whether the homotopy group is trivial. The author turns this into an algebraic question by showing that is the intersection of the terms of the lower central series of the crossed module , where is the 1-skeleton of .
The proof is based on the following algebraic result. Let and be totally free pre-crossed modules over the same group , and let be their coproduct (as pre-crossed -modules). Let and be the crossed modules induced by and . If the kernel of is trivial, then the kernel of is the intersection of the terms in the lower central series of .