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

AbstractSuppose 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 iX: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[iX∗:π1(X)→π1(M)]. We show that if G is not a cyclic group and M≇T2, K2 then E(X,M)0≃∗, if G is a nontrivial cyclic group and M≇P2, T2, K2 then E(X,M)0≃S1, 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 S1 is the circle, T2 is the torus, P2 is the projective plane and K2 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