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51
Spaces over a category and assembly maps in isomorphism conjectures
 in K  and Ltheory, K Theory 15
, 1998
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Continuoustrace algebras from the bundle theoretic point of view
 J. Austral. Math. Soc. Ser. A
, 1989
"... Using various facts about principal bundles over a space, we give a unified treatment of several theorems about the structure of stable separable continuoustrace algebras, their automorphisms, and their ATtheory. We also present a classification of real continuoustrace algebras from the same poin ..."
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Cited by 66 (7 self)
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Using various facts about principal bundles over a space, we give a unified treatment of several theorems about the structure of stable separable continuoustrace algebras, their automorphisms, and their ATtheory. We also present a classification of real continuoustrace algebras from the same point of view.
From loop groups to 2groups
 HHA
"... We describe an interesting relation between Lie 2algebras, the Kac– Moody central extensions of loop groups, and the group String(n). A Lie 2algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the ‘Jacobiator’. Similarly, a Lie 2gr ..."
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Cited by 49 (16 self)
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We describe an interesting relation between Lie 2algebras, the Kac– Moody central extensions of loop groups, and the group String(n). A Lie 2algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the ‘Jacobiator’. Similarly, a Lie 2group is a categorified version of a Lie group. If G is a simplyconnected compact simple Lie group, there is a 1parameter family of Lie 2algebras gk each having g as its Lie algebra of objects, but with a Jacobiator built from the canonical 3form on G. There appears to be no Lie 2group having gk as its Lie 2algebra, except when k = 0. Here, however, we construct for integral k an infinitedimensional Lie 2group PkG whose Lie 2algebra is equivalent to gk. The objects of PkG are based paths in G, while the automorphisms of any object form the levelk Kac– Moody central extension of the loop group ΩG. This 2group is closely related to the kth power of the canonical gerbe over G. Its nerve gives a topological group PkG  that is an extension of G by K(Z, 2). When k = ±1, PkG  can also be obtained by killing the third homotopy group of G. Thus, when G = Spin(n), PkG  is none other than String(n). 1 1
A Homology Theory for Étale Groupoids
"... Étale groupoids arise naturally as models for leaf spaces of foliations, for orbifolds, and for orbit spaces of discrete group actions. In this paper we introduce a sheaf homology theory for etale groupoids. We prove its invariance under Morita equivalence, as well as Verdier duality between Haeflig ..."
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Cited by 40 (6 self)
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Étale groupoids arise naturally as models for leaf spaces of foliations, for orbifolds, and for orbit spaces of discrete group actions. In this paper we introduce a sheaf homology theory for etale groupoids. We prove its invariance under Morita equivalence, as well as Verdier duality between Haefliger cohomology and this homology. We also discuss the relation to the cyclic and Hochschild homologies of Connes' convolution algebra of the groupoid, and derive some spectral sequences which serve as a tool for the computation of these homologies.
The polyhedral product functor: a method of computation for momentangle complexes, arrangements and related spaces
, 2008
"... This article gives a natural decomposition of the suspension of generalized momentangle complexes or partial product spaces which arise as polyhedral product functors described below. In the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the ..."
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Cited by 29 (7 self)
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This article gives a natural decomposition of the suspension of generalized momentangle complexes or partial product spaces which arise as polyhedral product functors described below. In the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the homological decomposition in GoreskyMacPherson [20], Hochster[22], Baskakov [3], Panov [36], and BuchstaberPanov [7]. Since the splitting is geometric, an analogous homological decomposition for a generalized momentangle complex applies for any homology theory. This decomposition gives an additive decomposition for the StanleyReisner ring of a finite simplicial complex and generalizations of certain homotopy theoretic results of Porter [39] and Ganea [19]. The spirit of the work here follows that of DenhamSuciu in [16].
Cyclic Cohomology of Étale Groupoids; The General Case
 Ktheory
, 1999
"... We give a general method for computing the cyclic cohomology of crossed products by 'etale groupoids, extending the FeiginTsyganNistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea, Connes, Feigin, Karoubi, Nistor and Tsygan for the convolution ..."
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Cited by 27 (1 self)
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We give a general method for computing the cyclic cohomology of crossed products by 'etale groupoids, extending the FeiginTsyganNistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea, Connes, Feigin, Karoubi, Nistor and Tsygan for the convolution algebra C 1 c (G) of an 'etale groupoid, removing the Hausdorffness condition and including the computation of hyperbolic components. Examples like group actions on manifolds and foliations are considered. Keywords: cyclic cohomology, groupoids, crossed products, duality, foliations. Contents 1 Introduction 3 2 Homology and Cohomology of Sheaves on ' Etale Groupoids 4 2.1 ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 \Gamma c in the nonHausdorff case : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.3 Homology and Cohomology of ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : 8 3 Cyclic Homologies of Sheaves ...
Simplicial cohomology with coefficients in symmetric categorical groups
, 2004
"... In this paper we introduce and study a cohomology theory {H n (−, A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group A a sequence of simplicial sets {K(A,n)}n≥0, which allows us to give a representation theorem for our cohomol ..."
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Cited by 23 (1 self)
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In this paper we introduce and study a cohomology theory {H n (−, A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group A a sequence of simplicial sets {K(A,n)}n≥0, which allows us to give a representation theorem for our cohomology. Moreover, we prove that for any n ≥ 3, the functor K(−,n) is right adjoint to the functor ℘n, where℘n(X•) is defined as the fundamental groupoid of the nloop complex � n (X•). Using this adjunction, we give another proof of how symmetric categorical groups model all homotopy types of spaces Y with πi(Y) = 0foralli = n, n + 1andn ≥ 3; and also we obtain a classification theorem for those spaces: [−,Y] ∼ = H n (−,℘n(Y)).
Grothendieck Groups of Poisson Vector Bundles
"... Abstract. A new invariant of Poisson manifolds, a Poisson Kring, is introduced. Hypothetically, this invariant is more tractable than such invariants as Poisson (co)homology. A version of this invariant is also defined for arbitrary algebroids. Basic properties of the Poisson Kring are proved and ..."
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Cited by 18 (0 self)
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Abstract. A new invariant of Poisson manifolds, a Poisson Kring, is introduced. Hypothetically, this invariant is more tractable than such invariants as Poisson (co)homology. A version of this invariant is also defined for arbitrary algebroids. Basic properties of the Poisson Kring are proved and the Poisson Krings are calculated for a number of examples. In particular, for the zero Poisson structure the Kring is the ordinary K 0ring of the manifold and for the dual space to a Lie algebra the Kring is the ring of virtual representations of the Lie algebra. It is also shown that the Kring is an invariant of Morita equivalence. Moreover, the Kring is a functor on a category, the weak