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On hypersphericity of manifolds with finite asymptotic dimension

Authors:	A N Dranishnikov

Institution:	Department of Mathematics, University of Florida, 358 Little Hall, P.O. Box 118105, Gainesville Florida 32611-8105

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 of bounded geometry is large scale uniformly embeddable into a Hilbert space, then is stably integrally hyperspherical.

Keywords:	Hyperspherical manifold uniform embedding asymptotic dimension scalar curvature Gromov-Lawson conjecture

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