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

Abstract

Suppose M is a connected PL 2-manifold and X is a compact connected subpolyhedron of M (X \neq 1pt, 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 \subset M in E(X, M). In this paper we classify the homotopy type of E(X, M)_0 in term of the subgroup G = Im[{i_X}_\ast : \pi_1(X) \to \pi_1(M)]. We show that if G is not a cyclic group and M \neq T^2, T^2 then E(X, M)_0 \simeq \ast, if G is a nontrivial cyclic group and M \neq P^2, T^2, K^2 then E(X, M)_0 \simeq S^1, and when G = 1, if X is an arc or M is orientable then E(X, M)_0 \simeq ST(M) and if X is not an arc and M is nonorientable then E(X, M)_0 \simeq ST(\tilde{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 \tilde{M} denotes the orientation double cover of M.