  We give a topological interpretation of the space of $L^2$-harmonic forms on
Manifold with flat ends. It is an answer to an old question of J. Dodziuk. We
also give a Chern-Gauss-Bonnet formula for the $L^2$-Euler characteristic of
some of these Manifolds. These results are applications of general theorems on
complete Riemannian Manifold whose Gauss-Bonnet operator is non-parabolic at
infinity.
