Multiplicative Structures on the Twisted Equivariant K-Theory of Finite Groups (original) (raw)
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Multiplicative Structures on the Twisted K-theory over Finite Groups
Let K be a finite group and let G be a finite group acting over K by automorphisms. In this paper we study two different but intimate related subjects: on the one side we classify all possible multiplicative and associative structures which one can endow the twisted G-equivariant K-theory over K, and on the other, we classify all possible monodical structures which one endow the category of twisted and G-equivariant bundles over K. We achieve this classification by encoding the relevant information in the cochains of a sub double complex of the double bar resolution associated to the semi-direct product K ⋊ G; we use known calculations of the cohomology of K, G and K ⋊ G to produce concrete examples of our classification. In the case on which K = G and G acts by conjugation, the multiplication map G ⋊ G → G is a homomorphism of groups and we define a shuffle homomorphism which realizes this map at the homological level. We show that the categorical information that defines the Twisted Drinfeld Double can be realized as the dual of the shuffle homomorphism applied to any 3-cocycle of G. We use the pullback of the multiplication map in cohomology to classify the possible ring structures that Grothendieck ring of representations of the Twisted Drinfeld Double may have, and we include concrete examples of this procedure.
Algebraic & Geometric Topology, 2014
We use a spectral sequence to compute twisted equivariant K-Theory groups for the classifying space of proper actions of discrete groups. We study a form of Poincaré Duality for twisted equivariant K-theory studied by Echterhoff, Emerson and Kim in the context of the Baum-Connes Conjecture with coefficients and verify it for the Group Sl 3 Z. In this work, we examine computational aspects relevant to the computation of twisted equivariant K-theory and K-homology groups for proper actions of discrete groups. Twisted K-theory was introduced by Donovan and Karoubi [DK70] assigning to a torsion element α ∈ H 3 (X, Z) abelian groups α K * (X) defined on a space by using finite dimensional matrix bundles. After the growing interest by physicists in the 1990s and 2000s, Atiyah and Segal [AS04] introduced a notion of twisted equivariant K-theory for actions of compact Lie Groups. In another direction, orbifold versions of twisted K-theory were introduced by Adem and Ruan [AR03], and progress was made to develop computational tools for Twisted Equivariant K-Theory with the construction of a spectral sequence in [BEUV13]. The paper [BEJU12] introduces Twisted equivariant K-theory for proper actions, allowing a more general class of twists, classified by the third integral Borel cohomology group H 3 (X × G EG, Z). We concentrate in the case of twistings given by discrete torsion, which is given by cocycles α ∈ Z 2 (G, S 1) representing classes in the image of the projection map H 2 (G, S 1) ∼ = → H 3 (BG, Z) → H 3 (X× G EG, Z). Under this assumption on the twist, a version of Bredon cohomology with coefficients in twisted representations can be used to approximate twisted equivariant K-Theory, by means of a spectral sequence studied in [BEUV13] and [Dwy08]. The Bredon (co)-homology groups relevant to the computation of twisted equivariant K-theory, and its homological version, twisted equivariant K-homology satisfy a Universal Coefficient Theorem, 1.13. We state it more generally for a pair of coefficient systems satisfying conditions 1.12. Theorem (Universal Coefficient Theorem). Let X be a proper, finite G-CW complex. Let M ? and M ? be a pair of functors satisfying Conditions 1.12. Then, there exists a short exact sequence of abelian groups 0 → Ext Z (H G n−1 (X, M ?), Z) → H n G (X, M ?) → Hom Z (H G n (X, M ?), Z) → 0
2004
Twisted complex K-theory can be defined for a space X equipped with a bundle of complex projective spaces, or, equivalently, with a bundle of C * -algebras. Up to equivalence, the twisting corresponds to an element of H 3 (X; Z). We give a systematic account of the definition and basic properties of the twisted theory, emphasizing some points where it behaves differently from ordinary K-theory. (We omit, however, its relations to classical cohomology, which we shall treat in a sequel.) We develop an equivariant version of the theory for the action of a compact Lie group, proving that then the twistings are classified by the equivariant cohomology group H 3 G (X; Z). We also consider some basic examples of twisted K-theory classes, related to those appearing in the recent work of Freed-Hopkins-Teleman.
Segal's Spectral Sequence in Twisted Equivariant K-theory for proper and discrete actions
We use the spectral sequence developed by Graeme Segal in order to understand the Twisted G-Equivariant K-Theory for proper and discrete actions. We show that the second page of this spectral sequence is isomorphic to a version of Equivariant Bredon cohomology with local coefficients in twisted representations. We furthermore give an explicit description of the third differential of the spectral sequence, and we recover known results when the twisting comes from finite order elements in discrete torsion. 1 2 NOÉ BÁRCENAS, JESÚS ESPINOZA, BERNARDO URIBE, AND MARIO VELÁSQUEZ
Twisted actions of categorical groups
We develop a theory of twisted actions of categorical groups using a notion of semidirect product of categories. We work through numerous examples to demonstrate the power of these notions. Turning to representations, which are actions that respect vector space structures, we establish an analog of Schur's lemma in this context. Keeping new terminology to a minimum, we concentrate on examples exploring the essential new notions introduced.
The Completion Theorem in twisted equivariant K-Theory for proper and discrete actions
We compare different algebraic structures in twisted equivariant K-Theory for proper actions of discrete groups. After the construction of a module structure over untwisted equivariant K-Theory, we prove a comple- tion Theorem of Atiyah-Segal type for twisted equivariant K-Theory. Using a Universal coefficient Theorem, we prove a cocompletion Theorem for Twisted Borel K-Homology for discrete Groups.
The twisted partial group algebra and (co)homology of partial crossed products
arXiv (Cornell University), 2023
Given a group G and a partial factor set σ of G, we introduce the twisted partial group algebra κ σ par G, which governs the partial projective σ-representations of G into algebras over a filed κ. Using the relation between partial projective representations and twisted partial actions we endow κ σ par G with the structure of a crossed product by a twisted partial action of G on a commutative subalgebra of κ σ par G. Then, we use twisted partial group algebras to obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the crossed product A * Θ G, involving the Hochschild homology of A and the partial homology of G, where Θ is a unital twisted partial action of G on a κ-algebra A with a κ-based twist. An analogous third quadrant cohomological spectral sequence is also obtained.