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• W2002400316 abstract "We show that for various natural classes of torsion-free groups, and appropriately defined Ktheoretic functors, the Isomorphism Conjecture is true if and only if a weaker Epimorphism Conjecture holds. Statement of results Let T F ⊂ (groups) denote the category of torsion-free discrete groups, with FL the full subcategory of groups G for which BG ≃ X a finite complex. For G ∈ obj(T F), the Baum-Connes Conjecture asserts that the classical assembly map KU∗(BG) → K t ∗(C ∗ r(G)) is an isomorphism, where KU∗(−) denotes complex K-homology, and K t (−) topological K-theory. More generally, for suitably defined functors F on the category (groups) of discrete groups, one has an assembly map HF∗(G) → F∗(G) and the Isomorphism Conjecture (IC) asserts that this map is an isomorphism, where HF∗(−) denotes the appropriate homology theory associated to F (for this note, we assume familiarity with IC; see [DL]). There are obvious variants on this conjecture. In particular, one can formulate an (apparantly weaker) Epimorphism Conjecture (EC), which states that the assembly map HF∗(G) → F∗(G) is only an epimorphism. Given a subring R ⊂ Q, the conjecture R-IC resp. R-EC is the conejcture that the assembly map is an isomorphism resp. epimorphism after tensoring with R. Finally, given a subcategory C ⊂ (groups), we say that R-IC or R-EC holds over C if the conjecture is true for all groups in C. Theorem 1. Let F∗(G) = K t (C ∗ r (G)) for a torsion-free discrete group G, with HF∗(G) := KU∗(BG). Then for all R ⊂ Q, R-EC holds for F over T F iff R-IC holds for F over T F. In particular, the BaumConnes Conjecture is true over this category iff the assembly map is an epimorphism for all torsion-free groups." @default.
• W2002400316 created "2016-06-24" @default.
• W2002400316 creator A5067101441 @default.
• W2002400316 date "2011-09-28" @default.
• W2002400316 modified "2023-09-27" @default.
• W2002400316 title "A remark on the Isomorphism Conjecture" @default.
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