For each $k \in \mathbb{Z}$, 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 $\mathbb{R}^n$, $n \ge 11$, 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 $K_*^{lf}(Z) \to K_*(C^*(Z))$ 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.
Alexander N. Dranishnikov 1 ; Steven C. Ferry 2 ; Shmuel Weinberger 3
@article{10_4007_annals_2003_157_919,
author = {Alexander N. Dranishnikov and Steven C. Ferry and Shmuel Weinberger},
title = {Large {Riemannian} manifolds which are flexible},
journal = {Annals of mathematics},
pages = {919--938},
year = {2003},
volume = {157},
number = {3},
doi = {10.4007/annals.2003.157.919},
mrnumber = {1983785},
zbl = {1051.53035},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.157.919/}
}
TY - JOUR AU - Alexander N. Dranishnikov AU - Steven C. Ferry AU - Shmuel Weinberger TI - Large Riemannian manifolds which are flexible JO - Annals of mathematics PY - 2003 SP - 919 EP - 938 VL - 157 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.157.919/ DO - 10.4007/annals.2003.157.919 LA - en ID - 10_4007_annals_2003_157_919 ER -
%0 Journal Article %A Alexander N. Dranishnikov %A Steven C. Ferry %A Shmuel Weinberger %T Large Riemannian manifolds which are flexible %J Annals of mathematics %D 2003 %P 919-938 %V 157 %N 3 %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.157.919/ %R 10.4007/annals.2003.157.919 %G en %F 10_4007_annals_2003_157_919
Alexander N. Dranishnikov; Steven C. Ferry; Shmuel Weinberger. Large Riemannian manifolds which are flexible. Annals of mathematics, Tome 157 (2003) no. 3, pp. 919-938. doi: 10.4007/annals.2003.157.919
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