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

TL;DR

This paper demonstrates that the homotopy type of the skeletal filtration of a CW-complex is invariant under cell decomposition changes, and introduces a new homotopy invariant based on crossed complexes and function spaces.

Contribution

It extends Whitehead's results by showing the independence of the homotopy type of the skeletal filtration from cell decomposition, and defines a new homotopy invariant using crossed complexes.

Findings

Homotopy type of skeletal filtration is cell decomposition independent.

Fundamental crossed complex depends only on the homotopy type of the space.

Introduces a homotopy invariant $I_A$ related to function spaces.

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 $\Pi(M)$ 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 $\Pi(D^n), n \in N$. This expands an old result (due to J.H.C. Whitehead) asserting that the homotopy type of $\Pi(M)$ 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 $A$.