ON FINITE INDUCED CROSSED MODULES AND THE HOMOTOPY TYPE OF MAPPING CONES Dedicated to the memory of J

Results on the niteness of induced crossed modules are proved both algebraically and topologically Using the Van Kampen type theorem for the fundamen tal crossed module applications are given to the types of mapping cones of classifying spaces of groups Calculations of the cohomology classes of some nite crossed modules are given using crossed complex methods Introduction Crossed modules were introduced by J H C Whitehead in They form a part of what can be seen as his programme of testing the idea of extending to higher dimensions the methods of combinatorial group theory of the s and of determining some of the extra structure that was necessary to model the geometry Other papers of Whitehead of this era show this extension of combinatorial group theory tested in di erent directions In this case he was concerned with the algebraic properties satis ed by the boundary map X A A of the second relative homotopy group together with the standard action on it of the fundamental group A This is the fundamental crossed module X A of the pair X A In order to determine the second homotopy group of a CW complex he formu lated and proved the following theorem for this structure Theorem W Let X A fe g be obtained from the connected space A by attaching cells Then the second relative homotopy group X A may be described as the free crossed A module on the cells The proof in uses transversality and knot theory ideas from the previous papers See for an exposition of this proof Several other proofs are available The The rst author was supported by EPSRC grant GR J for a visit to Zaragoza in November and is with Dr T Porter supported for equipment and software by EPSRC Grant GR J Non abelian homological algebra Received by the editors March and in revised form April Published on May Mathematics Subject Classi cation G F P Q