Bivariant K-theory and the Novikov conjecture

By N. Higson. 2000

134 citations recorded by the source corpus.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/69dc24c13e86d9f55b88451d7a513e474418dc78

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Citations in the corpus

Amenability and exactness for dynamical systems and their C*-algebras (2000, methodology): “This subject has attracted much attention these last three years in connection with the Novikov conjecture for discrete groups, following the works of Yu, Higson-Roe and Higson ([Yu],[HR],[ Hi ]).”
Banach KK-theory and the Baum-Connes conjecture (2003, background): “In the cases a), b), c) above, the proof of injectivity provides an explicit idempotent endomorphism on K*(C*ed(G)) whose image is the image of/ired (and the same for / i m a x and JJLL1)To state this we need to understand a baby case of Kasparov's if K-groups.”
The Novikov conjecture and geometry of Banach spaces (2012, methodology): “The idea of using the space of probability measures is due to Nigel Higson [7].”
Some notes on Property A (2006, methodology): “s a lot of the ideas in our discussion. The paper also removes the necessity of the ﬁniteness assumption on BGand the ﬁnite generation assumption on Gfor the main result of [Yu00], using an idea from [Hig00]. [Tes07] contains some applications of property A to the distortion of embeddings of certain”
Fibred coarse embeddings, a-T-menability and the coarse analogue of the Novikov conjecture (2013, unknown): “The primary application is to finitely generated discrete groups via the descent principle that is originally due to Higson [11], then adapted by Skandalis, Tu and Yu in [25]: Corollary 5.”
Noncommutative Poincare duality for boundary actions of hyperbolic groups (2003, background): “Each extension yields a map C(∂Γ)⋊Γ into the Calkin algebra, and these two maps into the Calkin algebra commute as a consequence of the compactification being, in the language of [13], ‘good,’ which simply means that metric balls of uniform size become small in the topology of the compactification n”
Boundary amenability of relatively hyperbolic groups (2005, methodology): “Besides those follow from exactness (e.g., the Novikov conjecture [Yu][ Hi ]), there are applications to the classification of orbit equivalence relations (this was the motivation of Adams [Ad]) as well as the classification of group von Neumann algebras.”
Amenable actions and exactness for discrete groups (2000, background): “Thus, there are n and unital completely positive maps φ: B(H) → Mn and ψ: Mn → B( H ) such that”
Geometry of the mapping class groups I: Boundary amenability (2005, background): “5 of [2]), as explained above the following corollary is immediate from Theorem 1 and the work of Higson [ 16 ].”
Metrics on diagram groups and uniform embeddings in a Hilbert space (2004, background): “That was proved in [28], see also [26], [19].”
The Novikov Conjecture for Linear Groups (2005, methodology): “Appealing to results of Skandalis, Tu and Yu [18,30] we conclude, as described in the Introduction, that the higher signatures associated to a homomorphism ρ : π1(M) → GL(n, K) are homotopy invariants.”
Going-down functors, the Künneth formula, and the Baum-Connes conjecture (2004, methodology): “…allowed to perform a two-step reduction of the problem: Step 1: Reduce the problem to the case of almost connected groups (based on a certain compression isomorphism and a Mayer-Vietoris argument — similar ideas have been used earlier very successfully for discrete groups in [17] [18] [28]).”
Geometrization of the Strong Novikov Conjecture for residually finite groups (2008, background): “The Strong Novikov Conjecture holds for many infinite groups with property T [5, 6, 9, 10, 14, 15, 24, 26, 27].”
The Baum-Connes and the Farrell-Jones Conjectures in K- and L-Theory (2004, methodology): “ch do not admit a uniform embedding into Hilbert space. Details of the construction are described in Ghys [134]. The next theorem is proved by Skandalis-Tu-Yu [289, Theorem 6.1] using ideas of Higson [148]. Theorem 5.9 (Injectivity of the Baum-Connes Assembly Map). Let Gbe a countable group. Suppose”