  Suppose M is a noncompact connected PL 2-manifold. In this paper we study the
topological property of the triple (H(M)_0, H^PL(M)_0, H^PL, c(M)_0), where
H(M)_0 is the identity component of the homeomorphism group {\cal H}(M) of M
with the compact-open topology, and H^PL(M)_0 and H^PL, c(M)_0 are the identity
components of the subgroups consisting of PL-homeomorphisms of M and ones with
compact supports. We show that this triple is a
(s^infty,sigma^infty,sigma^infty_f)-manifold and determine its topological
type. We also study the subgroups of Lipschitz homeomorphisms.
