TAILIEUCHUNG - Đề tài " Large Riemannian manifolds which are flexible "

For each k ∈ Z, we construct a uniformly contractible metric on Euclidean space which is not mod k hypereuclidean. We also construct a pair of uniformly contractible Riemannian metrics on Rn , n ≥ 11, so that the resulting manifolds Z and Z are bounded homotopy equivalent by a homotopy equivalence which is not boundedly close to a homeomorphism. We show that for these lf spaces the C ∗ -algebra assembly map K∗ (Z) → K∗ (C ∗ (Z)) from locally finite K-homology to the K-theory of the bounded propagation algebra is not a monomorphism . | Annals of Mathematics Large Riemannian manifolds which are flexible By A. N. Dranishnikov Steven C. Ferry and Shmuel Weinberger Annals of Mathematics 157 2003 919-938 Large Riemannian manifolds which are flexible By A. N. Dranishnikov Steven C. FERRy and Shmuel Weinberger Abstract For each k elL we construct a uniformly contractible metric on Euclidean space which is not mod k hypereuclidean. We also construct a pair of uniformly contractible Riemannian metrics on Rn n 11 so that the resulting manifolds Z and Z are bounded homotopy equivalent by a homotopy equivalence which is not boundedly close to a homeomorphism. We show that for these spaces the C -algebra assembly map k J Z K C Z from locally finite K-homology to the K-theory of the bounded propagation algebra is not a monomorphism. This shows that an integral version of the coarse Novikov conjecture fails for real operator algebras. If we allow a single cone-like singularity a similar construction yields a counterexample for complex C -algebras. 1. Introduction This paper is a contribution to the collection of problems that surrounds positive scalar curvature topological rigidity . the Borel conjecture the Novikov and Baum-Connes conjectures. Much work in this area see . 14 4 3 15 has focused attention on bounded and controlled analogues of these problems which analogues often imply the originals. Recently success in attacks on the Novikov and Gromov-Lawson conjectures has been achieved along these lines by proving the coarse Baum-Connes conjecture for certain classes of groups 23 27 28 . A form of the coarse Baum-Connes conjecture states that the C -algebra assembly map ịi K X K C X is an isomorphism for uniformly contractible metric spaces X with bounded geometry 21 . Using work of Gromov on embedding of expanding graphs in groups r with Br a finite complex 16 the epimorphism part of the coarse Baum-Connes con The authors are partially supported by NSF grants. The second author would like to thank .

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