Asphericity and approximation properties of crossed modules

This paper is devoted to the study of the Baer invariants and approximation properties of crossed modules and $\text{cat}^1$-groups. Conditions are considered under which the kernels of crossed modules coincide with the intersection of the lower central series. An algebraic criterion for asphericity is produced for two-dimensional complexes having aspherical plus-construction. As a consequence it is shown that a subcomplex of an aspherical two-dimensional complex is aspherical if and only if its fundamental $\text{cat}^1$-group is residually soluble. Thus, a new formulation in group-theoretic terms is given to the Whitehead asphericity conjecture.
Bibliography: 25 titles.