Math 2.0

K-theory Higher algebraic K-theory Conjecture

Farrell–Jones conjecture

Builds on Assembly mapGroup ringK-theory spectrumL-theoryGroup actionRing with involution

Statement

Let GG be a group and let RR be a ring. Let E𝒱𝒞GE_{\mathcal{VC}}G be a classifying space for the family of virtually cyclic subgroups of GG. That is, E𝒱𝒞GE_{\mathcal{VC}}G is a GG-CW complex whose fixed point set (E𝒱𝒞G)H(E_{\mathcal{VC}}G)^H is contractible when H≤GH \le G is virtually cyclic and empty otherwise. Write HnG(−;𝐊R)H^G_n(-; \mathbf{K}_R) for the GG-equivariant homology theory of Davis and Lück with coefficients in algebraic K-theory of RR.

  1. (K-theory) The assembly map HnG(E𝒱𝒞G;𝐊R)→HnG(pt;𝐊R)=Kn(R[G])H^G_n(E_{\mathcal{VC}}G; \mathbf{K}_R) \to H^G_n(\mathrm{pt}; \mathbf{K}_R) = K_n(R[G]) is an isomorphism for every n∈ℤn \in \Z.
  2. (L-theory) If RR is a ring with involution, the assembly map HnG(E𝒱𝒞G;𝐋R⟨−∞⟩)→Ln⟨−∞⟩(R[G])H^G_n(E_{\mathcal{VC}}G; \mathbf{L}^{\langle -\infty \rangle}_R) \to L^{\langle -\infty \rangle}_n(R[G]) in L-theory is an isomorphism for every n∈ℤn \in \Z.

Remarks

Farrell and Jones stated the conjecture in their 1993 paper on isomorphism conjectures in algebraic K-theory. It is open in general. It is known for large classes of groups, including hyperbolic groups and CAT(0) groups, by work of Bartels, Lück, Reich and Wegner. For torsion-free GG and regular RR, the K-theory part reduces to the classical assembly map Hn(BG;K(R))→Kn(R[G])H_n(BG; K(R)) \to K_n(R[G]) being an isomorphism. For torsion-free GG it implies that the Whitehead group Wh(G)Wh(G), the reduced group K~0(ℤ[G])\widetilde{K}_0(\Z[G]) and the negative K-groups of ℤ[G]\Z[G] vanish. The L-theory part implies the Novikov conjecture for GG. The two parts together imply the Borel conjecture for closed aspherical manifolds of dimension at least 55 with fundamental group GG. The K-theory part implies the Kaplansky idempotent conjecture for torsion-free GG and fields of characteristic zero.

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Assembly map, concept, no evidence levelAssembly mapconceptGroup action, concept, no evidence levelGroup actionconceptGroup ring, concept, no evidence levelGroup ringconceptK-theory spectrum, concept, no evidence levelK-theory spectrumconceptL-theory, concept, no evidence levelL-theoryconceptRing with involution, concept, no evidence levelRing with involutionconceptFarrell–Jones conjecture, conjecture, no evidence levelFarrell–Jones conject…conjecture
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  1. Assembly map concept
  2. Group action concept
  3. Group ring concept
  4. K-theory spectrum concept
  5. L-theory concept
  6. Ring with involution concept
  7. Farrell–Jones conjecture conjecture

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Further reading

  1. Web https://en.wikipedia.org/wiki/Farrell%E2%80%93Jones_conjecture
About this page
TagB2X59B7J8V
URIm2://B2X59B7J8V
Kindconjecture
WritingW0 stub
MSC19D50, 19G24, 57R67
StatusThis copy of the site has no build data. Run m2 status B2X59B7J8V.
AboutAssembly map, Group ring
UsesAssembly map, Group ring, K-theory spectrum, L-theory, Group action, Ring with involution
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Changed3 November 2026

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