Two results on asphericity

Abstract

We construct an explicit finite chain of non-aspherical group presentation complexes $K(\mathcal{X})\subset K(\mathcal{Y})\subset K(\mathcal{Z})$ in which $K(\mathcal{Y})$ is Cockcroft, both inclusion-induced maps on ${\pi }_2$ are zero, and the pair $(K(\mathcal{Y}),K(\mathcal{X}))$ 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.