[Paper Review] On the Farrell-Jones and related Conjectures

Wolfgang Lueck|ArXiv.org|Oct 11, 2007

Homotopy and Cohomology in Algebraic Topology121 references3 citations

TL;DR

This paper provides a comprehensive introduction to the Farrell-Jones and Baum-Connes Conjectures, framing them as powerful tools that relate difficult-to-compute algebraic and topological K-theory groups to more tractable equivariant homology theories via assembly maps. It establishes their structural and computational significance, proving them for broad classes of groups and highlighting their implications for longstanding conjectures in topology and K-theory.

ABSTRACT

These extended notes are based on a series of six lectures presented at the summer school ``Cohomology of groups and algebraic $K$-theory'' which took place in Hangzhou, China from July 1 until July 12 in 2007. They give an introduction to the Farrell-Jones and the Baum-Connes Conjecture.

Motivation & Objective

• To provide a self-contained introduction to the Farrell-Jones and Baum-Connes Conjectures for researchers in algebraic topology and K-theory.
• To clarify the role of assembly maps in relating complex K- and L-theory groups to computable equivariant homology theories.
• To demonstrate how these conjectures unify and imply major classical conjectures such as those of Bass, Borel, Kaplansky, and Novikov.
• To survey known proofs and open cases, emphasizing the current state of the art and unresolved challenges in the field.

Proposed method

• Utilizes homological algebra and CW-complexes to define and analyze projective modules and K-groups, particularly $K_0$, $K_1$, and negative $K$-theory.
• Applies the Bass-Heller-Swan decomposition to express $K_*(RG)$ in terms of $K_*(R)$ and $K_*(R[G])$ for group rings.
• Introduces classifying spaces for families of subgroups to construct the target of the assembly map in the Farrell-Jones Conjecture.
• Employs equivariant homology theories as the framework for the assembly maps in both the Farrell-Jones and Baum-Connes settings.
• Leverages controlled topology techniques—especially forget-control maps and geometric constructions like contracting maps—for proving surjectivity in the Farrell-Jones Conjecture.
• Reviews analytic methods such as the Dirac-Dual-Dirac method in $KK$-theory for the Baum-Connes Conjecture, particularly via Kasparov's bivariant $KK$-theory.

Experimental results

Research questions

• RQ1 How do the Farrell-Jones and Baum-Connes Conjectures unify and generalize classical conjectures in algebraic topology and K-theory?
• RQ2 What is the role of assembly maps in connecting complex K- and L-theory groups to more accessible equivariant homology theories?
• RQ3 For which classes of groups have the Farrell-Jones and Baum-Connes Conjectures been proven, and what are the key methods used in these proofs?
• RQ4 Why is the Baum-Connes Conjecture considered more suspicious than the Farrell-Jones Conjecture, particularly regarding potential counterexamples?
• RQ5 What are the limitations of analytic methods in proving the Farrell-Jones Conjecture, and why is a transfer of techniques from controlled topology to $KK$-theory not yet realized?

Key findings

• The Farrell-Jones Conjecture has been proven for hyperbolic groups, CAT(0)-groups, and solvable groups, with the $K$-theoretic version known for $SL_n(\mathbb{Z})$ when $n \leq 2$.
• The Baum-Connes Conjecture is known to hold for hyperbolic groups, $SL_n(\mathbb{Z})$ for $n \leq 2$, and all amenable groups, with the $K$-theory of the reduced group $C^*$-algebra being computable in these cases.
• The conjectures imply the Bost Conjecture, the Kaplansky Zero Divisor Conjecture, and the Novikov Conjecture, demonstrating their foundational role in geometric topology.
• For Thompson’s group $F$, injectivity of the assembly map holds after inverting 2 for $R = \mathbb{Z}$, though the full conjecture remains open.
• The version of the Baum-Connes Conjecture with coefficients fails for certain groups with expanders, as shown by Higson, Lafforgue, and Skandalis, indicating a potential source of counterexamples.
• Despite extensive investigation, no group is currently known to be a counterexample to the Farrell-Jones or Baum-Connes Conjectures, though such candidates remain elusive.