On the homotopy type and the fundamental crossed complex of the skeletal filtration of a CW-complex (Q869726)

The author first presents as ''preliminaries'' some results on cofibred filtrations, crossed complexes (extensively studied and used by J. H. C. Whitehead, H. J. Baues, R. Brown and P. J. Higgins) and then interesting original results on ``The homotopy type of the skeletal filtration of a CW complex'' and ''On the fundamental crossed complex of a CW-complex''. He proves 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 \(D^{n}\) , when the latter are given with their natural CW-decomposition with unique cells of order 0, \(n-1\) and \(n\). So, \(\prod (M)\) depends only on the homotopy type of M, as a space, up to free products with crossed complexes of the type \(D^{n}=\prod (D^{n})\), where \(n\in N\). This theorem is deduced (making use of the Higher Homotopy van Kampem Theorem) from an analogous statement on CW-complexes proved in this article : The homotopy type of a CW-complex M as a filtered space depends only on the homotopy type of M, up to wedge products with CW-complexes of the type \( D^{n},n\in N\), provided with their natural cell decompositions. This result expands an old result due to J.H.C. Whitehead asserting that the homotopy type of \(\prod (M)\) depends only on the homotopy type of M. These results are used to define a homotopy invariant \(I_{A}\) of CW-complexes for each finite crossed complex A. It is interpreted in terms of the weak homotopy type of the function space \(Top((M,\ast ),(| A| ,\ast ))\) where \(| A| \) is the classifying space of the crossed complex A.