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

Abstract

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 $D^n$, 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 $Π(D^n), n \in N$. 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 $I_A$ 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 $A$.