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Large Riemannian manifolds which are flexible. (English) Zbl 1051.53035

The first major result of the paper is the theorem that for any given \(k\) and \(n\geq 8\) there is a Riemannian manifold \(Z\) diffeomorphic to \(\mathbb R^n\) such that \(Z\) is uniformly contractible, but admits no coarsely proper coarse Lipschitz map \(Z\to\mathbb R^n\) of degree 1 or of degree \(\equiv 1 \mod k\) (though such a map of some nonzero integral degree does exist).
The construction is based on examples (due to the first author) of compact metric spaces with infinite covering dimension and finite (equal to 3) cohomological dimension, which solved in negative an Alexandrov problem, or in another setting, on examples of cell-like maps \(M\to X\) from closed manifolds onto a compact space which can raise the dimension.
A similar construction combined with bounded surgery theory provides examples of \(Z\), \(Z'\) diffeomorphic to \(\mathbb R^n\) for any \(n\geq 11\) with a coarse isometry and hence a bounded homotopy equivalence between them, which is however not boundedly homotopic to a homeomorphism. This is the second major result of the paper.
Finally, using the same construction the authors provide examples of uniformly contractible singular Riemannian manifolds \(\overline M\) for which the assembly map \[ K_{\ast}^{\text{lf}}(\overline M)\to K_{\ast}(C^{\ast}(\overline M)) \]
from locally finite \(K\)-homology of \(M\) to the \(K\)-theory of the bounded propagation algebra is not a monomorphism. This shows that a coarse form of the integral Novikov conjecture fails for uniformly contractible manifolds without the bounded geometry condition.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C20 Global Riemannian geometry, including pinching
57R65 Surgery and handlebodies
57N60 Cellularity in topological manifolds
19K35 Kasparov theory (\(KK\)-theory)
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