Sobolev homeomorphisms are dense in volume preserving automorphisms

In this paper we prove a weak version of Lusin's theorem for the space of Sobolev-$(1,p)$ volume preserving homeomorphisms on closed and connected n-dimensional manifolds, $n\ge 3$, for $p<n?1$. We also prove that if $p>n$ this result is not true. More precisely, we obtain the density of Sobolev-$(1,p)$ homeomorphisms in the space of volume preserving automorphisms, for the weak topology. Furthermore, the regularization of an automorphism in a uniform ball centered at the identity can be done in a Sobolev-$(1,p)$ ball with the same radius centered at the identity.