We use a 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 Bredon cohomology with local coefficients in twisted representations. We furthermore explain some phenomena concerning 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 VELÁSQUEZFor this purpose we develop a twisted version of Bredon cohomology, cohomology which turns out to determine the E 2 -page of Segal's spectral sequence once it is applied to an equivariantly contractible cover.The construction of the spectral sequence extends and generalizes previous work of C. Dwyer [12], who only treated the twistings which are classified by cohomology classes of finite order which lie in the image of the canonical map H 3 (BG, Z) → H 3 (X× G EG; Z); these twistings take the name of discrete torsion twistings.The main result of this note, which is Theorem 4.7, relies on the construction and the properties of the universal stable equivariant projective unitary bundle carried out in [6]. Since this work can be seen as a continuation of what has been done in [6], we will use the notation, the definitions and the results of that paper. We will not reproduce any proof that already appears in [6], instead we will give appropriate references whenever a definition or a result of [6] is used.We emphasize that the topological issues that may appear when working with the Projective Unitary Group have all been resolved in [22, Section 15] when it is endowed with the norm topology. We therefore assume in this work that we are working with the norm topology when discussing topological properties of operator spaces.This note is organized as follows. In Section 1 a version of Bredon cohomology associated to an equivariant cover of a space is constructed. In Section 2 the basics of Transformation Groups and Parametrized Homotopy Theory needed for the construction are quickly reviewed. This is used to construct a version of Bredon cohomology with local coefficients. In Section 3 the construction of twisted equivariant K-theory for proper and discrete actions given in [6] is reviewed. In Section 4 , the Bredon cohomology with local coefficients in twisted representations is shown to be isomorphic to the second page of a spectral sequence converging to twisted equivariant K-theory. Some phenomena concerning the third differential of this spectral sequence is also analyzed. In Section 5 some simple examples are given including the case of discrete torsion which was developed by Dwyer in [12].Acknowledgements.

Abstract. Following ideas of Graeme Segal, we construct an equivariant configuration space that is a model of equivariant connective K-homology spectrum for finite groups, as a consequence we obtain an induction structure for equivariant connective K-homology. We describe explicitly the homology with complex coefficients for the fixed points of this configuration space as a Hopf algebra.

We compare twisted equivariant K-theory of SL 3 Z with untwisted equivariant K-theory of a central extension St 3 Z. We compute all twisted equivariant K-theory groups of SL 3 Z, and compare them with previous work on the equivariant K-theory of BSt 3 Z by Tezuka and Yagita.Using a universal coefficient theorem by the authors, the computations explained here give the domain of Baum-Connes assembly maps landing on the topological K-theory of twisted group C * -algebras related to SL 3 Z, for which a version of KKtheoretic duality studied by Echterhoff, Emerson, and Kim is verified.The second author was financially supported under the project Aplicaciones de la K-teoría en teoría delíndice y las conjeturas de isomorfismo with ID

Using a combination of Atiyah-Segal ideas on one side and of Connes and BaumConnes ideas on the other, we prove that the Twisted geometric K-homology groups of a Lie groupoid have an external multiplicative structure extending hence the external product structures for proper cases considered by Adem-Ruan in [1] or by Tu,Xu and Laurent-Gengoux in [24]. These Twisted geometric K-homology groups are the left hand sides of the twisted geometric Baum-Connes assembly maps recently constructed in [9] and hence one can transfer the multiplicative structure via the Baum-Connes map to the Twisted K-theory groups whenever this assembly maps are isomorphisms.

We develop a spectral sequence of Eilenberg-Moore type to compute Bredon Cohomology of spaces with an action of a group given as a pullback.Using several other spectral sequences, and positive results on the Baum-Connes Conjecture, we are able to compute Equivariant K-Theory and K-homology of the reduced C * -algebra of a 6-dimensional crystallographic group Γ introduced by Vafa and Witten. We also use positive results on the Farrell-Jones conjecture to give a vanishing result for the algebraic K-theory of the group ring of the group Γ in negative degrees.