Universal twist in equivariant K -theory for proper and discrete actions

Abstract

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.

Equivariant principal bundles and their classifying spaces

“…Then a Γ-equivariant principle PU(H)-bundle p : E → B whose family of local representations is contained in S is the same as a Γ-equivariant stable projective unitary bundle over B in the sense of [5,Definition 2.2]. Note that the existence of the local data appearing in [5, Definition 2.2] is automatically satisfied by Theorem 9.1.…”

“…namely, as the homotopy groups of the space of Γ-equivariant sections of the associated Fred(H)-bundle; see [5,Appendix A] for further properties of the twisted equivariant K-theory groups defined in this way. Now, in the case that Γ is furthermore a Lie group we can show that there is an isomorphism between the set Bundle Γ st (X, PU(H)) and H 3 (EΓ × Γ X, Z).…”

Twisted geometric K-homology for proper actions of discrete groups

“…In section 2, we recall definitions, methods, and results related to twisted equivariant K-Theory for proper actions [5], [4], [3].…”

“…In section 3, geometric cycles for twisted equivariant K-homology are introduced, as well as appropriate equivalence relations (bordism, isomorphism and vector bundle modification). It is proved in Theorem 3.9, that the geometric Twisted Equivariant K-Homology groups satisfy dual homological properties to the twisted equivariant K-Theory groups in [5]. In that section computational methods, including a spectral sequence abutting to Twisted Equivariant K-Homology are addressed.…”

“…The author benefited from conversations and joint work with Mario Velásquez and Paulo Carrillo. Twisted Equivariant K-Theory for proper actions of discrete groups was introduced in [5]. We will recall definitions, results and methods from [5] and [3].…”