The Baum-Connes Conjecture and Coarse Geometry

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

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