Equivariant K-theory of compact Lie groups with involution (original) (raw)

Proper actions and decompositions in equivariant K-theory

arXiv: Algebraic Topology, 2020

In this paper we study a natural decomposition of GGG-equivariant KKK-theory of a proper GGG-space, when GGG is a Lie group with a compact normal subgroup AAA acting trivially. Our decomposition could be understood as a generalization of the theory known as Mackey machine under suitable hypotheses, since it decomposes GGG-equivariant K-theory in terms of twisted equivariant K-theory groups respect to some subgroups of G/AG/AG/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 KKK-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 KKK-homology for actions of compact Lie groups using a suitable configuration space model, based on previous papers published by the third a...

Equivariant geometric K-homology for compact Lie group actions

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 2010

Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups K G * (X), using an obvious equivariant version of the (M, E, f )-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the "official" equivariant K-homology groups) and show that these are isomorphisms.

Twisted equivariant K-theory with complex coefficients

Journal of Topology, 2007

Using a global version of the equivariant Chern character, we describe the complexified twisted equivariant K-theory of a space with a compact Lie group action in terms of fixed-point data. We apply this to the case of a compact group acting on itself by conjugation, and relate the result to the Verlinde algebra and to the Kac numerator at q = 1. Verlinde's formula is also discussed in this context.

A Fixed Point Decomposition of Twisted Equivariant K-Theory

2022

We present a decomposition of rational twisted G-equivariant Ktheory, G a finite group, into cyclic group equivariant K-theory groups of fixed point spaces. This generalises the untwisted decomposition by Atiyah and Segal [AS89] as well as the decomposition by Adem and Ruan for twists coming from group cocycles [AR03].

Multiplicative Structures on the Twisted Equivariant K-Theory of Finite Groups

Let K be a finite group and let G be a finite group acting on K by automorphisms. In this paper we study two different but intimately related subjects: on the one side we classify all possible multiplicative and associative structures with which one can endow the twisted G-equivariant K-theory of K, and on the other, we classify all possible monoidal structures with which one can 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 semidirect 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 in 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 the Grothendieck ring of representations of the Twisted Drinfeld Double may have, and we include concrete examples of this procedure.