[Paper Review] Twisted equivariant K-theory for proper actions of discrete groups

Christopher Dwyer|ArXiv.org|Oct 10, 2007

Homotopy and Cohomology in Algebraic Topology10 references3 citations

TL;DR

This paper constructs twisted equivariant K-theory for proper actions of discrete groups using projective representations and group cohomology, extending Adem and Ruan's finite group framework via Lück and Oliver's techniques. The key contribution is a spectral sequence that collapses at the $E_2$ page after tensoring with $\mathbb{C}$, establishing a natural link to the non-twisted Atiyah-Hirzebruch spectral sequence and providing a computational tool for torsion twistings.

ABSTRACT

We give a construction for twisted equivariant K-theory in the case of a proper action of a discrete group using twisted bundles. Our construction uses results of Lueck and Oliver to extend a construction of Adem and Ruan. We also show the existence of a Chern character to twisted Bredon cohomology. This gives a partial answer to the question of when you can construct twisted equivariant K-theory out of finite rank twisted bundles.

Motivation & Objective

• To extend twisted equivariant K-theory to proper actions of discrete groups beyond finite groups.
• To generalize Adem and Ruan's construction for finite groups using Lück and Oliver's methods.
• To establish a spectral sequence relating twisted Bredon cohomology to twisted equivariant K-theory.
• To show that the spectral sequence collapses at $E_2$ after tensoring with $\mathbb{C}$, linking it to the non-twisted Atiyah-Hirzebruch spectral sequence.
• To provide a computational framework for torsion twistings in the context of orbifolds and groupoid $C^*$-algebras.

Proposed method

• Uses $\alpha$-twisted representations of finite groups to model twisted K-theory, where $\alpha$ is a 2-cocycle in $Z^2(G, \mathbb{C}^*)$.
• Constructs the $\alpha$-twisted representation group $R_\alpha(G)$ as the Grothendieck group of $\alpha$-twisted representations.
• Introduces a twisted product on coefficient functors analogous to the one in Adem and Ruan's construction.
• Defines twisted Bredon cohomology with values in the $\alpha$-twisted representation group and establishes its Mackey functor structure after tensoring with $\mathbb{Q}$.
• Applies a result of Lück to show that the spectral sequence from twisted Bredon cohomology to twisted equivariant K-theory collapses at the $E_2$ page after tensoring with $\mathbb{C}$.
• Uses the equivariant Chern character to identify the $E_2$ and $E_\infty$ pages after $\mathbb{Q}$-tensor, confirming the collapse.

Experimental results

Research questions

• RQ1 Can twisted equivariant K-theory be extended from finite groups to proper actions of general discrete groups?
• RQ2 How does the spectral sequence relating twisted Bredon cohomology to twisted equivariant K-theory behave, and does it collapse?
• RQ3 What is the relationship between the twisted spectral sequence and the non-twisted Atiyah-Hirzebruch spectral sequence?
• RQ4 Can torsion twistings arising from group cohomology be realized via finite rank twisted bundles?
• RQ5 How does this construction relate to the broader framework of twisted K-theory in orbifold and operator algebra settings?

Key findings

• The spectral sequence from twisted Bredon cohomology to twisted equivariant K-theory collapses at the $E_2$ page after tensoring with $\mathbb{C}$.
• The $E_2$ page of the spectral sequence is isomorphic to the $E_\infty$ page after $\mathbb{C}$-tensor, due to the equivariant Chern character being an isomorphism in this setting.
• The spectral sequence is a module over the non-twisted equivariant Atiyah-Hirzebruch spectral sequence in a natural way.
• The construction provides a finite rank $\alpha$-twisted bundle for any torsion twisting $[\alpha] \in H^3(X \times_G EG; \mathbb{Z})$ that lies in the image of $p^* \circ \delta$.
• The Chern character in this framework offers an alternative computational method for twisted K-theory groups in the torsion case, distinct from the operator algebraic approach of Tu and Xu.
• The result supports the conjecture in [22] that torsion twistings may admit finite rank twisted bundles, providing a partial positive answer.