Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems

Brown, Ronald

doi:10.1090/fic/043/05

Cited by 16 publications

(32 citation statements)

References 76 publications

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“…We refer, for example to [1,2,5,8,9,15,14,20]. A natural generalisation of the concept of a crossed module is a crossed complex: Definition 2.10.…”

Section: Crossed Complexesmentioning

confidence: 99%

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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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“…We refer, for example to [1,2,5,8,9,15,14,20]. A natural generalisation of the concept of a crossed module is a crossed complex: Definition 2.10.…”

Section: Crossed Complexesmentioning

confidence: 99%

“…Crossed complexes are studied or used extensively in [1,2,5,8,10,11,12,13,14,38,39], for example. Notice that H.J.…”

Section: Blakers In [3]mentioning

confidence: 99%

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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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“…This is a crossed complex of free type, called the "Fundamental Crossed Complex of M "; see for example [5,6]. In particular, we have an action of the group π 1 (M 1 , * ) on all the other groups, preserving the boundary maps, and such that, if n > 2, then the action of π 1 …”

Section: Theorem 18 the Natural Morphism From The Free Crossed Modumentioning

confidence: 99%

The fundamental crossed module of the complement of a knotted surface

Martins

2009

Trans. Amer. Math. Soc.

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Abstract. We prove that if M is a CW-complex and M 1 is its 1-skeleton, then the crossed module Π 2 (M, M 1 ) depends only on the homotopy type of M as a space, up to free products, in the category of crossed modules, with Π 2 (D 2 , S 1 ). From this it follows that if G is a finite crossed module and M is finite, then the number of crossed module morphisms Π 2 (M, M 1 ) → G can be re-scaled to a homotopy invariant I G (M ), depending only on the algebraic 2-type of M . We describe an algorithm for calculating π 2 (M, M (1) ) as a crossed module over π 1 (M (1) ), in the case when M is the complement of a knotted surface Σ in S 4 and M (1) is the handlebody of a handle decomposition of M made from its 0-and 1-handles. Here, Σ is presented by a knot with bands. This in particular gives us a geometric method for calculating the algebraic 2-type of the complement of a knotted surface from a hyperbolic splitting of it. We prove in addition that the invariant I G yields a non-trivial invariant of knotted surfaces in S 4 with good properties with regard to explicit calculations.

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“…Analogous ideas in the globular case present conceptual and technical difficulties. Multiple compositions allow easily an algebraic inverse to subdivision, and this is a key to certain local-to-global results, as outlined for example in [6]. However the algebraic relations between the cubical, globular, and (in the groupoid case) crossed complexes, are an essential part of the picture.…”

Section: Higher Dimensionsmentioning

confidence: 99%

Three themes in the work of Charles Ehresmann: local-to-global; groupoids; higher dimensions

Brown

2007

Banach Center Publications

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Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems

Cited by 16 publications

References 76 publications

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

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

The fundamental crossed module of the complement of a knotted surface

Three themes in the work of Charles Ehresmann: local-to-global; groupoids; higher dimensions

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