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# Homology, Homotopy and Applications

## Volume 9 (2007)

### Number 1

### On the homotopy type and the fundamental crossed complex of the skeletal filtration of a CW-complex

Pages: 295 – 329

DOI: https://dx.doi.org/10.4310/HHA.2007.v9.n1.a13

#### Author

#### 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 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 $\Pi(M)$ 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 ${\cal D}^n \doteq \Pi(D^n)$, $n \in \mathbb{N}$. 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 $I_{\mathcal{A}}$ of CW-complexes for each finite crossed complex ${\mathcal{A}}$. We interpret it in terms of the weak homotopy type of the function space $\mathit{TOP} \big ((M,*),(|{\mathcal{A}}|,*)\big)$, where $|{\mathcal{A}}|$ is the classifying space of the crossed complex ${\mathcal{A}}$.

#### Keywords

CW-complex, skeletal filtration, crossed complex, higher homotopy van Kampen Theorem, invariants of homotopy types

#### 2010 Mathematics Subject Classification

55P10, 55Q05, 57M27

Published 1 January 2007