Spaces of embeddings of compact polyhedra into 2-manifolds

Let M be a PL 2-manifold and X be a compact subpolyhedron of M and let E(X,M) denote the space of embeddings of X into M with the compact-open topology. In this paper we study an extension property of embeddings of X into M and show that the restriction map from the homeomorphism group of M to E(X,M) is a principal bundle. As an application we show that if M is a Euclidean PL 2 -manifold and dim X≥1 then the triple (E(X,M) , ELIP(X,M) , EPL(X,M)) is an (s,Σ,σ) -manifold, where EKLIP(X,M) and EKPL(X,M) denote the subspaces of Lipschitz and PL embeddings.