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 Dn, when the latter 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 higher homotopy van Kampen Theorem (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 Π(Dn), 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 each 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.
Homology, Homotopy and Applications, Vol. 9 (2007), No. 1, pp.295-329.
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