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Roughly speaking coarse space is a set X together with a family of subsets of the product of X with itself, called enourages, which play the role of the sets of pairs of points in a metric space of distance R or less apart, where R is a natural number. In a coarse space one can speak of a family of pairs of points being uniformly boundedly separated, but one does not have available a notion of convergence: there is no small-scale structure to a course space, only large-scale structure. It is possible to associate to a coarse geometric space a C*-algebra - this construction is due mostly to John Roe. The Baum-Connes conjecture for coarse geometric spaces proposes a formula for the K-theory of this C*-algebra. The formula is inspired by closely related conjectures in controlled topology. A key feature of the coarse Baum-Connes conjecture is that in many cases the Baum-Connes conjecture for the coarse space underlying a discrete group implies the injectivity of the Baum-Connes assembly map for the group itself. This in turn has powerful topological consequences like the Novikov conjecture. This has led to a great deal of progress on the Novikov conjecture, including most notably Yu's theorem that the Novikov conjecture holds for groups which admit a uniform embedding into Hilbert space |
Selected References |
| Dranishnikov, A. N. Asymptotic topology. (Russian) Uspekhi Mat. Nauk 55 (2000), no. 6(336), 71-116; translation in Russian Math. Surveys 55 (2000), no. 6, 1085-1129 MR |
| Higson, Nigel; Roe, John. On the coarse Baum-Connes conjecture. Novikov conjectures, index theorems and rigidity, Vol. 2 (Oberwolfach, 1993), 227-254, London Math. Soc. Lecture Note Ser., 227, Cambridge Univ. Press, Cambridge, 1995. MR |
| Roe, John. Index theory, coarse geometry, and topology of manifolds. CBMS Regional Conference Series in Mathematics, 90. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. x+100 pp. ISBN: 0-8218-0413-8 MR |
| Roe, John. Lectures on coarse geometry. University Lecture Series, 31. American Mathematical Society, Providence, RI, 2003. viii+175 pp. ISBN: 0-8218-3332-4 MR |
| Roe, John. Warped cones and property A. Geom. Topol. 9 (2005), 163-178 (electronic). MR |
| Skandalis, G.; Tu, J. L.; Yu, G. The coarse Baum-Connes conjecture and groupoids. Topology 41 (2002), no. 4, 807-834. MR |
| Tu, Jean-Louis. The gamma element for groups which admit a uniform embedding into Hilbert space. Recent advances in operator theory, operator algebras, and their applications, 271-286, Oper. Theory Adv. Appl., 153, Birkhaeuser, Basel, 2005. MR |
| Yu, Guo Liang. Baum-Connes conjecture and coarse geometry. K-Theory 9 (1995), no. 3, 223-231. MR |
| Yu, Guo Liang. Coarse Baum-Connes conjecture. K-Theory 9 (1995), no. 3, 199-221. MR |
| Yu, Guoliang. Localization algebras and the coarse Baum-Connes conjecture. K-Theory 11 (1997), no. 4, 307-318. MR |
| Yu, Guoliang. The Novikov conjecture for groups with finite asymptotic dimension. Ann. of Math. (2) 147 (1998), no. 2, 325-355. MR |
| Yu, Guoliang. The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math. 139 (2000), no. 1, 201-240. MR |