Twisted equivariant K-theory with complex coefficients

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

Algebraic and Geometric Topology, 2014

Replaces Previous version. Includes comments on poincare duality for twisted equivariant in the context of proper and discrete actions and the Baum-Connes Conjecture. We use a spectral sequence proposed by C. Dwyer and previous work by Sanchez-Garcia and Soule to compute Twisted Equivariant K-theory groups of the classifying space for proper actions of Sl3(Z). After proving a Universal coefficient theorem in Bredon Cohomology with specific coefficients, we compute the twisted equivariant K-homology and state a relation to the Baum-Connes Conjecture with coefficients.

Journal of Homotopy and Related Structures, 2022

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 completion 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.

K-Theory and Noncommutative Geometry, 2008

Twisted K-theory has its origins in the author's PhD thesis [27] and in a paper with P. Donovan [19]. The objective of this paper is to revisit the subject in the light of new developments inspired by Mathematical Physics. See for instance E. Witten [42], J. Rosenberg [37], C. Laurent-Gentoux, J.-L. Tu, P. Xu [41] and M.F. Atiyah, G. Segal [8], among many authors. We also prove some new results in the subject: a Thom isomorphism, explicit computations in the equivariant case and new cohomology operations.

Proceedings of the London Mathematical Society, 2013

We define equivariant projective unitary stable bundles as the appropriate twists when defining K-theory as sections of bundles with fibers the space of Fredholm operators over a Hilbert space. We construct universal equivariant projective unitary stable bundles for the orbit types, and we use a specific model for these local universal spaces in order to glue them to obtain a universal equivariant projective unitary stable bundle for discrete and proper actions. We determine the homotopy type of the universal equivariant projective unitary stable bundle, and we show that the isomorphism classes of equivariant projective unitary stable bundles are classified by the third equivariant integral cohomology group. The results contained in this paper extend and generalize results of Atiyah-Segal.

Cohomological Methods in Homotopy Theory, 2001

In an earlier paper [10], we showed that for any discrete group G, equivariant K-theory for finite proper G-CW-complexes can be defined using equivariant vector bundles. This was then used to prove a version of the Atiyah-Segal completion theorem in this situation. In this paper, we continue to restrict attention to actions of discrete groups, and begin by constructing an appropriate classifying space which allows us to define K * G (X) for an arbitrary proper G-complex X. We then construct rational-valued equivariant Chern characters for such spaces, and use them to prove some more general versions of completion theorems. In fact, we construct two different types of equivariant Chern character, both of which involve Bredon cohomology with coefficients in the system G/H → R(H). The first, ch

2005

Let G be a finite group acting on a finite dimensional real vector space V. We denote by P(V) the projective space associated to V. In this paper we compute in a very explicit way the rank of the equivariant complex K-theory of V and P(V), using previous results by Atiyah and the author. The interest of this computation comes from explicit formulas given by the Baum-Connes-Slominska Chern character and the basic fact that the equivariant K-theory of V is free. We use these topological computations to prove algebraic results like computing the number of conjugacy classes of G which split in a central extension. Our main example is the case where V = R^n and G = the symmetric group of n letters acting on V by permutation of the coordinates. This example is related to the famous pentagonal identity of Euler and (ironically) the Euler-Poincare characteristic of the equivariant K-theory of V.

Publications Mathématiques de l'IHÉS, 1968

Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2014

For a compact simply connected simple Lie group G with an involution α, we compute the G ⋊ ℤ/2-equivariant K-theory of G where G acts by conjugation and ℤ/2 acts either by α or by g ↦ α(g)−1. We also give a representation-theoretic interpretation of those groups, as well as of KG(G).

arXiv: Algebraic Topology, 2020

In this paper we study a natural decomposition of $G$-equivariant $K$-theory of a proper $G$-space, when $G$ is a Lie group with a compact normal subgroup $A$ acting trivially. Our decomposition could be understood as a generalization of the theory known as Mackey machine under suitable hypotheses, since it decomposes $G$-equivariant K-theory in terms of twisted equivariant K-theory groups respect to some subgroups of $G/A$. Similar decompositions were known for the case of a compact Lie group acting on a space, but our main result applies to discrete, linear and almost connected groups. We also apply this decomposition to study equivariant $K$-theory of spaces with only one isotropy type. We provide a rich class of examples in order to expose the strength and generality of our results. We also study the decomposition for equivariant connective $K$-homology for actions of compact Lie groups using a suitable configuration space model, based on previous papers published by the third a...

2003

This is the third paper of a series relating the equivariant twisted $K$-theory of a compact Lie group $G$ to the ``Verlinde space'' of isomorphism classes of projective lowest-weight representations of the loop groups. Here, we treat arbitrary compact Lie groups. In addition, we discuss the relation to semi-infinite cohomology, the fusion product of Conformal Field theory, the r\^ole of energy and the topological Peter-Weyl theorem.

2008

In this paper, we develop differential twisted K-theory and define a twisted Chern character on twisted K-theory which depends on a degree three 3 Deligne cocycle. We also establish the general Riemann-Roch theorem in twisted K-theory and find some applications in the study of twisted K-theory of compact simple Lie groups.

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

Given a C * -algebra A with a KMS weight for a circle action, we construct and compute a secondary invariant on the equivariant K-theory of the mapping cone of A T → A, both in terms of equivariant KK-theory and in terms of a semifinite spectral flow. This in particular puts the previously considered examples of Cuntz algebras [9] and SU q (2) [13] in a general framework. As a new example we consider the Araki-Woods III λ representations of the Fermion algebra.

We construct two new G-equivariant rings: K(X, G), called the stringy K-theory of the G-variety X, and H(X, G), called the stringy cohomology of the G-variety X, for any smooth, projective variety X with an action of a finite group G. For a smooth Deligne-Mumford stack X, we also construct a new ring K orb (X) called the full orbifold K-theory of X. We show that for a global quotient X = [X/G], the ring of G-invariants K orb (X) of K(X, G) is a subalgebra of K orb ([X/G]) and is linearly isomorphic to the "orbifold K-theory" of Adem-Ruan [AR] (and hence Atiyah-Segal), but carries a different "quantum" product which respects the natural group grading. We prove that there is a ring isomorphism Ch : K(X, G) → H(X, G), which we call the stringy Chern character. We also show that there is a ring homomorphism Ch orb : K orb (X) → H • orb (X), which we call the orbifold Chern character, which induces an isomorphism Ch orb : K orb (X) → H • orb (X) when restricted to the sub-algebra K orb (X). Here H • orb (X) is the Chen-Ruan orbifold cohomology. We further show that Ch and Ch orb preserve many properties of these algebras and satisfy the Grothendieck-Riemann-Roch theorem with respect toétale maps. All of these results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds.

Journal of Geometry and Physics, 2009

In this paper, we develop differential twisted K-theory and define a twisted Chern character on twisted K-theory which depends on a choice of connection and curving on the twisting gerbe. We also establish the general Riemann-Roch theorem in twisted K-theory and find some applications in the study of twisted K-theory of compact simple Lie groups.