HyperEuclidean manifolds and the Novikov Conjecture

TL;DR

This paper introduces new Lipschitz homotopy techniques for manifolds with finite asymptotic dimension, proving the Novikov conjecture for certain groups and defining asymptotically piecewise Euclidean spaces.

Contribution

It develops novel methods for analyzing manifolds with finite asymptotic dimension and proves the Novikov conjecture for these groups using alternative approaches.

Findings

Higson compactification is mod p acyclic for finite-dimensional, uniformly contractible manifolds.

Higher Signature Novikov Conjecture holds for groups with finite asymptotic dimension.

Expanders are not asymptotically piecewise Euclidean.

Abstract

We develop some basic Lipschitz homotopy technique and apply it to manifolds with finite asymptotic dimension. In particular we show that the Higson compactification of a uniformly contractible manifold is mod $p$ acyclic in the finite dimensional case. Then we give an alternative proof of the Higher Signature Novikov Conjecture for the groups with finite asymptotic dimension. Finally we define an asymptotically piecewise Euclidean metric space as a space which admits an approximation by Euclidean asymptotic polyhedra. We show that the Gromov-Lawson conjecture holds for the asymptotically piecewise Euclidean groups. Also we prove that expanders are not asymptotically piecewise Euclidean