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

Martins, João Faria

doi:10.4310/hha.2007.v9.n1.a13

Cited by 8 publications

(31 citation statements)

References 35 publications

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“…The results of this subsection extend in a natural way for the case of the fundamental crossed complex of a CW-complex; see [18].…”

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

confidence: 80%

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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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“…The results of this subsection extend in a natural way for the case of the fundamental crossed complex of a CW-complex; see [18].…”

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

confidence: 80%

“…The results in this subsection extend in a natural way to crossed complexes; see [18]. Let M be a path-connected topological space.…”

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confidence: 93%

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The fundamental crossed module of the complement of a knotted surface

Martins

2009

Trans. Amer. Math. Soc.

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“…. , N } → (P i , γ i , θ i ), where P i ∈ L 2 , γ i is a quantised path from v to x Pi , and (27) it follows:…”

Section: Algebraic Topology Preliminaries For the 2-sphere Casementioning

confidence: 99%

Higher lattices, discrete two-dimensional holonomy and topological phases in (3 + 1)D with higher gauge symmetry

et al. 2019

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Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we will continue the study of Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. In particular, we show that a previously proposed construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in 3+1 dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly combinatorialised CWdecompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group.The operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretised 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.1 Due to a lack of local observables, experimentally distinguishing different topological phases can be a difficult task from a microscopic point of view. Instead the characterising properties of topological phases are most efficiently described by their emergent behaviours. Signatures for the presence of topological order include ground state degeneracies which depend on the spatial topology of the material in question [69,43], universal negative corrections to the entanglement entropy [44,37,48] and the presence of stable topological excitations which provide non-trivial representations of their respective motion groups [47,41]. This means the braid group for point particles (anyons) in 2+1D [54] and the loop braid group for loop excitations in 3+1D [66].In 2+1D there exist several constructions for TQFTs (see for instance [64]). Path-integral models arise from Chern-Simons-Witten theory [73] and from BF theory [1], while the discrete realisation of BF-theory coincides with the Turaev-Viro [64]/Barrett-Westbury [6] state-sum (see [2]). In contrast, in 3+1D a framework general enough to capture all features of 4D topology is still lacking. Nevertheless, we have the Crane-Yetter TQFT [23,64] and its generalisations [71,25]; and the Yetter homotopy 2-type TQFT [74,59,33], derived from a strict finite 2-group [4]. All of these 3+1D TQFTs give rise to topological invariant...

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“…In this way we get the classifying space of a crossed complex [BH91]. Its properties are further exploited in, for example, [BGPT,FMa07,FMP06]. However the exposition in [BHS08] adopts an earlier cubical approach.…”

Section: Topological Datamentioning

confidence: 99%

A new higher homotopy groupoid: the fundamental globular $ω$-groupoid of a filtered space

Brown

2008

Homology, Homotopy and Applications

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We show that the graded set of filter homotopy classes rel vertices of maps from the n-globe to a filtered space may be given the structure of (strict) globular ω-groupoid. The proofs use an analogous fundamental cubical ω-groupoid due to the author and Philip Higgins in 1981. This method also relates the construction to the fundamental crossed complex of a filtered space, and this relation allows the proof that the crossed complex associated to the free globular ω-groupoid on one element of dimension n is the fundamental crossed complex of the n-globe.

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

Cited by 8 publications

References 35 publications

The fundamental crossed module of the complement of a knotted surface

The fundamental crossed module of the complement of a knotted surface

Higher lattices, discrete two-dimensional holonomy and topological phases in (3 + 1)D with higher gauge symmetry

A new higher homotopy groupoid: the fundamental globular $ω$-groupoid of a filtered space

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