Homotopy types of the components of spaces of embeddings of compact polyhedra into 2-manifolds

Suppose M is a connected PL 2-manifold and X is a compact connected subpolyhedron of M ( X ≠ 1 pt, a closed 2-manifold). Let E ( X , M ) denote the space of topological embeddings of X into M with the compact-open topology and let E ( X , M ) 0 denote the connected component of the inclusion i X : X ⊂ M in E ( X , M ) . In this paper we classify the homotopy type of E ( X , M ) 0 in terms of the subgroup G = Im [ i X ∗ : π 1 ( X ) → π 1 ( M ) ] . We show that if G is not a cyclic group and M ≇ T 2 , K 2 then E ( X , M ) 0 ≃ ∗ , if G is a nontrivial cyclic group and M ≇ P 2 , T 2 , K 2 then E ( X , M ) 0 ≃ S 1 , and when G = 1 , if X is an arc or M is orientable then E ( X , M ) 0 ≃ ST ( M ) and if X is not an arc and M is nonorientable then E ( X , M ) 0 ≃ ST ( M ˜ ) . Here S 1 is the circle, T 2 is the torus, P 2 is the projective plane and K 2 is the Klein bottle. The symbol ST ( M ) denotes the tangent unit circle bundle of M with respect to any Riemannian metric of M and when M is nonorientable, M ˜ denotes the orientable double cover of M.