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In mathematics, the Farrell–Jones conjecture,〔Farrell, F.T., Jones, L.E., Isomorphism conjectures in algebraic K-theory, ''J. Amer. Math. Soc.'', v. 6, pp. 249–297, 1993〕 named after F. Thomas Farrell (now at (SUNY Binghamton )) and Lowell Edwin Jones (now at (SUNY Stony Brook )) states that certain assembly maps are isomorphisms. These maps are given as certain homomorphisms. The motivation is the interest in the target of the assembly maps; this may be, for instance, the algebraic K-theory of a group ring : or the L-theory of a group ring :, where ''G'' is some group. The sources of the assembly maps are equivariant homology theory evaluated on the classifying space of ''G'' with respect to the family of virtually cyclic subgroups of ''G''. So assuming the Farrell–Jones conjecture is true, it is possible to restrict computations to virtually cyclic subgroups to get information on complicated objects such as or . The Baum-Connes conjecture formulates a similar statement, for the topological K-theory of reduced group -algebras . ==Formulation== One can find for any ring equivariant homology theories satisfying : respectively Here denotes the group ring. The K-theoretic Farrell–Jones conjecture for a group ''G'' states that the map induces an isomorphism on homology : Here denotes the classifying space of the group ''G'' with respect to the family of virtually cyclic subgroups, i.e. a ''G''-CW-complex whose isotropy groups are virtually cyclic and for any virtually cyclic subgroup of ''G'' the fixed point set is contractible. The L-theoretic Farrell–Jones conjecture is analogous. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Farrell–Jones conjecture」の詳細全文を読む スポンサード リンク
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