Homology, Homotopy and Applications

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

Joã0 Faria Martins

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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 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 II$(M)$ 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 $\mathcal{D}^n \doteq \Pi (D^n), n \in \mathbb{N}$.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 $TOP ((M, *), (|\mathcal{A}|, *))$ where $|\mathcal{A}|$ is the classifying space of the crossed complex $\mathcal{A}$.

Article information

Source
Homology Homotopy Appl., Volume 9, Number 1 (2007), 295-329.

Dates
First available in Project Euclid: 5 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.hha/1175791098

Mathematical Reviews number (MathSciNet)
MR2299802

Zentralblatt MATH identifier
1114.55002

Subjects
Primary: 55P10: Homotopy equivalences 55Q05: Homotopy groups, general; sets of homotopy classes 57M27: Invariants of knots and 3-manifolds

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

Citation

Martins, Joã0 Faria. On the homotopy type and the fundamental crossed complex of the skeletal filtration of a CW-complex. Homology Homotopy Appl. 9 (2007), no. 1, 295--329. https://projecteuclid.org/euclid.hha/1175791098


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