  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
