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On hypersphericity of manifolds with finite asymptotic dimension

Author: A. N. Dranishnikov
Journal: Trans. Amer. Math. Soc. 355 (2003), 155-167
MSC (2000): Primary 53C23
Published electronically: September 6, 2002
MathSciNet review: 1928082
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Abstract: We prove the following embedding theorems in the coarse geometry:
\begin{theorem1}Every metric space $X$\space with bounded geometry whose asympt... ... embedding into the product of $n+1$\space locally finite trees. \end{theorem1}

\begin{theorem2}Every metric space $X$\space with bounded geometry whose asympt... ...ding into a non-positively curved manifold of dimension $2n+2$ . \end{theorem2}
The Corollary is used in the proof of the following.
\begin{theorem3}For every uniformly contractible manifold $X$ whose asymptotic d... ...\mathbf{R}^{n}$\space is integrally hyperspherical for some $n$ . \end{theorem3}

Theorem B together with a theorem of Gromov-Lawson implies the result, previously proven by G. Yu (1998), which states that an aspherical manifold whose fundamental group has a finite asymptotic dimension cannot carry a metric of positive scalar curvature.

We also prove that if a uniformly contractible manifold $X$ of bounded geometry is large scale uniformly embeddable into a Hilbert space, then $X$ is stably integrally hyperspherical.

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Additional Information

A. N. Dranishnikov
Affiliation: Department of Mathematics, University of Florida, 358 Little Hall, P.O. Box 118105, Gainesville Florida 32611-8105

Keywords: Hyperspherical manifold, uniform embedding, asymptotic dimension, scalar curvature, Gromov-Lawson conjecture
Received by editor(s): January 23, 2001
Received by editor(s) in revised form: May 20, 2002
Published electronically: September 6, 2002
Additional Notes: The author was partially supported by NSF grant DMS-9971709
Article copyright: © Copyright 2002 American Mathematical Society