  Suppose M is a noncompact connected smooth 2-manifold without boundary and
let D(M)_0 denote the identity component of the diffeomorphism group of M with
the compact-open C^infty-topology. In this paper we investigate the topological
type of D(M)_0 and show that D(M)_0 is a topological ell_2-manifold and it has
the homotopy type of the circle if M is the plane, the open annulus or the open
M"obius band, and it is contractible in all other cases. When M admits a volume
form w, we also discuss the topological type of the group of w-preserving
diffeomorphisms of M. To obtain these results we study some fundamental
properties of transformation groups on noncompact spaces endowed with weak
topology.
