On the Homotopy Type and the Fundamental Crossed Complex of the Skeletal Filtration of a CW-Complex

We prove that if M is a CW-complex, then the homotopy type of the skeletal filtration of M does not depend on the cell decomposition of M up to wedge products with n-disks D, when the later are given their natural CW-decomposition with unique cells of order 0, (n − 1) and n; a result resembling J.H.C. Whitehead’s work on simple homotopy types. From the Colimit Theorem for the Fundamental Crossed Complex of a CW-complex (due to R. Brown and P.J. Higgins), follows an algebraic analogue for the fundamental crossed complex Π(M) of the skeletal filtration of M , which thus depends only on the homotopy type of M (as a space) up to free product with crossed complexes of the type D . = Π(D), n ∈ N. This expands an old result (due to J.H.C. Whitehead) asserting that the homotopy type of Π(M) depends only on the homotopy type of M . We use these results to define a homotopy invariant IA of CW-complexes for any finite crossed complex A. We interpret it in terms of the weak homotopy type of the function space TOP ((M, ∗), (|A|, ∗)), where |A| is the classifying space of the crossed complex A. 2000 Mathematics Subject Classification: 55P10, 55Q05, 57M05.