Lecture Notes in Algebraic Topology

Davis, James F.; Kirk, Paul

doi:10.1090/gsm/035

Cited by 196 publications

(213 citation statements)

References 23 publications

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“…✷ Given a finite CW -complex X, the Atiyah-Hirzebruch spectral sequence [2] for the topological K-theory of X is a particular case of Serre spectral sequence [25,50] for the trivial fibration of X by itself with fiber a point:…”

Section: Proof Of Theoremmentioning

confidence: 99%

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A spectral sequence for theK-theory of tiling spaces

Savinien

Bellissard

2009

Ergod. Th. Dynam. Sys.

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Let T be an aperiodic and repetitive tiling of R d with finite local complexity. We present a spectral sequence that converges to the K-theory of T with page-2 given by a new cohomology that will be called PV in reference to the Pimsner-Voiculescu exact sequence. It is a generalization of the Serre spectral sequence. The PV cohomology of T generalizes the cohomology of the base space of a fibration with local coefficients in the K-theory of its fiber. We prove that it is isomorphic to thě Cech cohomology of the hull of T (a compactification of the family of its translates). Main ResultsLet T be an aperiodic and repetitive tiling of R d with finite local complexity (definition 3). The hull Ω is a compactification, with respect to an appropriate topology, of the family of translates of T by vectors of R d (definition 4). The tiles of T are given compatible ∆-complex decompositions (section 4.2), with each simplex punctured, and the ∆-transversal Ξ ∆ is the subset of Ω corresponding to translates of T having the puncture of one of those simplices at the origin 0 R d . The prototile space B 0 (definition 9) is built out of the prototiles of T (translational equivalence classes of tiles) by gluing them together according to the local configurations of their representatives in the tiling. The hull is given a dynamical system structure via the natural action of the group R d on itself by translation [13]. The C * -algebra of the hull is isomorphic to the crossed-productThere is a map p o from the hull onto the prototile space (proposition 1)which, thanks to a lamination structure on Ω (remark 2), resembles (although is not) a fibration with base space B 0 and fiber Ξ ∆ (remark 4). The Pimsner-Voiculescu (PV) *

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Section: Proof Of Theoremmentioning

confidence: 99%

“…The original reference on exact couples is the work of Massey [49]. The link with spectral sequences is only mentioned, and the reader is referred to [25,50] for further details. An exact couple is a family T = (D, E, i, j, k) where D and E are abelian groups and i, j, k are group homomorphisms making the following triangle exact…”

Section: Propositionmentioning

confidence: 99%

A spectral sequence for theK-theory of tiling spaces

Savinien

Bellissard

2009

Ergod. Th. Dynam. Sys.

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“…This corresponds to regarding any left C[π 1 (U)]-module A as a right C[π 1 (U)]-module by setting a·γ = γ −1 ·a, for all a ∈ A and γ ∈ π 1 (U), and extending by linearity. Following [11], p. 97, we regard in this paper C 0 * = C * ⊗ C as a complex of right R s -modules.…”

Section: −1mentioning

confidence: 99%

Multivariable Alexander invariants of hypersurface complements

Dimca

Maxim

2007

Trans. Amer. Math. Soc.

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Abstract. We start with a discussion on Alexander invariants, and then prove some general results concerning the divisibility of the Alexander polynomials and the supports of the Alexander modules, via Artin's vanishing theorem for perverse sheaves. We conclude with explicit computations of twisted cohomology following an idea already exploited in the hyperplane arrangement case, which combines the degeneration of the Hodge to de Rham spectral sequence with the purity of some cohomology groups.

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“…Universal coefficient theorems, among other things, provide a measure of how this adjoint fails to be an isomorphism in terms of Ext and T or [10]. Here q represents the dimension of the space for which the (co)homology is calculated.…”

Section: Independence Of Topology and The Universal Coefficient Theoremmentioning

confidence: 99%

Black holes, information, and the universal coefficient theorem

Patrascu

2016

Journal of Mathematical Physics

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General relativity is based on the diffeomorphism covariant formulation of the laws of physics while quantum mechanics is based on the principle of unitary evolution. In this article I provide a possible answer to the black hole information paradox by means of homological algebra and pairings generated by the universal coefficient theorem. The unitarity of processes involving black holes is restored by demanding invariance of the laws of physics to the change of coefficient structures in cohomology.

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Lecture Notes in Algebraic Topology

Cited by 196 publications

References 23 publications

A spectral sequence for theK-theory of tiling spaces

A spectral sequence for theK-theory of tiling spaces

Multivariable Alexander invariants of hypersurface complements

Black holes, information, and the universal coefficient theorem

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