The L-theory Farrell–Jones conjecture for torsionfree groups

From papers

Let GG be a group, let RR be a ring with involution, and let L(R)\mathbf{L}^{\langle-\infty\rangle}(R) be the associated LL-theory spectrum, whose nnth homotopy group is Ln(R)L_n^{\langle-\infty\rangle}(R). The L-theory Farrell–Jones conjecture. If GG is torsionfree, then the assembly map

Hn(BG;L(R))Ln(RG)H_n(BG;\mathbf{L}^{\langle-\infty\rangle}(R))\longrightarrow L_n^{\langle-\infty\rangle}(RG)

is an isomorphism for every nZn\in\mathbb{Z}. This is the LL-theoretic counterpart of the algebraic KK-theory Farrell–Jones conjecture; the source gives no resolution status.

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Sources & referencesView supporting material

Primary source

Wolfgang Lueck, “Survey on the Farrell-Jones Conjecture”, arXiv:2507.11337 (2025).

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