On hypersphericity of manifolds with finite asymptotic dimension

Abstract:

We prove the following embedding theorems in the coarse geometry:

Theorem A. Every metric space $X$ with bounded geometry whose 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 whose 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 B For every uniformly contractible manifold $X$ whose asymptotic dimension is finite, the product $X\times \mathbf {R}^{n}$ 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.