On the Isomorphism Conjecture in algebraic K -theory

By Arthur Bartels, T. Farrell, L. Jones, H. Reich. 2001

68 citations recorded by the source corpus.

The Isomorphism Conjecture is a conceptional approach towards a calculation of the algebraic K-theory of a group ring RΓ, where Γ is an infinite group. In this paper we prove the conjecture in dimensions n<2 for fundamental groups of closed Riemannian manifolds with strictly negative sectional curvature and arbitrary coefficient rings R. If R is regular this leads to a concrete calculation of low dimensional K-theory groups of RΓ in terms of the K-theory of R and the homology of the group.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e824a63a28a8e623cdbab7b9d81838d382c6f2b5

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In Geometry and Topology

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Citations in the corpus

The Baum-Connes and the Farrell-Jones Conjectures in K- and L-Theory (2004, unknown): “It extends Bartels–Farrell-Jones–Reich [19].”
Coefficients for the Farrell¿Jones Conjecture (2005, methodology): “If a set E of subsets of X ◊X and a set F of subsets of X satisfies the conditions (i)-(iv) listed in [ BFJR04 , Subsection 2.3] then we speak of E respectively F as morphism and object support conditions and define”
Identifying assembly maps in K- and L-theory (2004, background): “On the other hand, the category BG(Z, X; R) is not itself a fixed-point category in any obvious way, as required for the applications to the Novikov conjecture in [CP95], [CPV98]. The following lemma is proved in [ BFJR ]”
The K-theoretic Farrell–Jones conjecture for hyperbolic groups (2007, background): “The first result dealing with arbitrary coefficient rings R appear in Bartels–Farrell–Jones– Reich [3], where the K -theoretic Farrell–Jones conjecture was proven in dimension ≤ 1 for G the fundamental group of a negatively curved closed Riemannian manifold.”
The Borel Conjecture for hyperbolic and CAT(0)-groups (2009, background): “The Farrell-Jones Conjectures have been generalized to include group rings RG over arbitrary rings [4], and further to twisted group rings which are best treated using the language of actions of G on additive categories [5], [11].”
Finite decomposition complexity and the integral Novikov conjecture for higher algebraic K-theory (2011, background): “Since X has bounded geometry, the path γ: [0, 1] → Ps(X) meets only finitely many (closed) simplices σ1, .”