Embedding spaces and hyperspaces of polyhedra in 2-manifolds

AbstractSuppose M is a 2-manifold and X is a compact polyhedron. Let E(X,M) denote the space of embeddings of X into M with the compact-open topology and let K(X,M) denote the hyperspace of copies of X embedded in M with the Fréchet topology. In this paper we show that the natural map π:E(X,M)→K(X,M), π(f)=f(X), is a principal bundle with fiber H(X), the homeomorphism group of X. As a corollary it follows that the space K(X,M) is an ANR