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-Jones 1993 citation
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 section
The class of groups which satisfies the fibered Farrell-Jones conjecture contain the following groups: virtually cyclic groups (definition), hyperbolic groups, CAT(0) groups, solvable groups, mapping class groups.
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