Computing crossed modules induced by an
inclusion of a normal subgroup, with applications to homotopy
2-types

Ronald Brown   and Christopher D. Wensley
School of Mathematics
University of Wales, Bangor
Gwynedd LL57 1UT, U.K.


We obtain some explicit determinations of 
crossed Q-modules induced from a crossed module over a normal
subgroup P of Q. By virtue of theorems of Brown and Higgins,
this  enables the  computation of the homotopy 2-types, and 
so of the second homotopy modules, of
homotopy pushouts of maps of classifying spaces of
discrete groups induced by certain morphisms of the groups. In some cases, 
the first k-invariant is calculated, using methods of free 
crossed resolutions.