We construct an explicit finite chain of non-aspherical group presentation complexes K(X)⊂K(Y)⊂K(Z) in which K(Y) is Cockcroft, both inclusion-induced maps on π2
Nearby in the stack
are zero, and the pair
(K(Y),K(X))
has the identity property. This answers a question of Hitchman. The construction is given directly from a short balanced presentation of the binary icosahedral group. As the second result, we construct an integral Lie ring presentation in which a subpresentation has a nontrivial identity among relations, whereas the presentation itself has none.