On hypersphericity of manifolds with finite asymptotic dimension

We prove the following embedding theorems in the coarse geometry: Theorem A. Every metric space X with bounded geometry hose asymptotic dimension does not exceed n admits a large scale uniform embedding into the product of n + 1 locally finite trees. Corollary. Every metric space X with bounded geometry hose asymptotic dimension does not exceed n admits a large scale uniform embedding into a non-positively curved manifold of dimension 2n + 2. The Corollary is used in the proof of the following. Theorem. For every uniformly contractible manifold X hose asymptotic dimension is finite, the product X x R-n is integrally hyperspherical for some n. 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.