The Farrell-Jones conjecture, named after F. Thomas Farrell and Lowell E. Jones, states in mathematics that certain assembly maps relating a group's equivariant homology to the algebraic K-theory or L-theory of its group ring are isomorphisms. The conjecture is motivated by interest in the target of these assembly maps, since the algebraic K-theory and L-theory of a group ring are often difficult to compute directly. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementThe Farrell-Jones conjecture, named after F. Thomas Farrell and Lowell E. Jones, states that certain assembly maps are isomorphisms. 1 Proposed YearYear of the Farrell and Jones paper introducing the conjecture. Progress Toward ResolutionThe fibered Farrell-Jones conjecture is proven for several classes of groups, including virtually cyclic, hyperbolic, CAT(0), solvable and mapping class groups; the general conjecture remains open. 1 Classification
Resolution Status Prize Status
Prize Status (category) Sources
1. Farrell-Jones Conjecture (Wikipedia)
Wikimedia FoundationLead section
the Farrell-Jones conjecture, named after F. Thomas Farrell and Lowell E. Jones, states that certain assembly maps are isomorphisms
Lead section, full first two sentences
In mathematics, the Farrell-Jones conjecture, named after F. Thomas Farrell and Lowell E. Jones, states that certain assembly maps are isomorphisms. These maps are given as certain homomorphisms.
References
Farrell, F. Thomas, Jones, Lowell E., Isomorphism conjectures in algebraic K-theory, Journal of the American Mathematical Society, v. 6, pp. 249-297, 1993
Inheritances of isomorphism conjectures
The class of groups which satisfies the fibered Farrell-Jones conjecture contain the following groups virtually cyclic groups (definition) hyperbolic groups (see ) CAT(0) groups (see ) solvable groups (see ) mapping class groups (see ) Furthermore the class has the following inheritance properties: Closed under finite products of groups.
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