Some Relations Between Homology and Homotopy Groups

Blakers, A. L.

doi:10.2307/1969290

Cited by 47 publications

(27 citation statements)

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“…It generalises the concept of the classifying space of a group. A similar construction of classifying spaces of crossed complexes appeared in [3].…”

Section: The Classifying Space Of a Crossed Complexmentioning

confidence: 90%

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

Martins

2007

Homology, Homotopy and Applications

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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 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 typeThis 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 each 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.

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“…It generalises the concept of the classifying space of a group. A similar construction of classifying spaces of crossed complexes appeared in [3].…”

Section: The Classifying Space Of a Crossed Complexmentioning

confidence: 90%

“…3 Since M is a subcomplex of N , and N is homotopic to M , it follows that M is a strong deformation retract of N . By the Cellular Approximation Theorem, we can suppose, furthermore, that there exists a retraction r :…”

Section: Dimension Twomentioning

confidence: 99%