Embedding spaces and hyperspaces of polyhedra in 2-manifolds
Let \(M\) be a 2-manifold and \(X\) a compact polyhedron. Denote by \({\mathcal E}(X,M)\) the space of embeddings of \(X\) into \(M\) and by \({\mathcal H}(X)\) the group of homeomorphisms of \(X\), both spaces equipped with the compact-open topology. Finally, denote by \({\mathcal K}(X,M)\) the hyperspace consisting of the subsets of \(M\) homeomorphic to \(X\). Equip \({\mathcal K}(X,M)\) with the Fréchet topology, a metric topology defined by considering distances between subsets of \(M\) homeomorphic to \(X\). It is proved that the natural map \(\pi: {\mathcal E}(X,M) \to {\mathcal K}(X,M)\), defined by \({\pi}(f) = f(X)\), is a principal bundle with fibre \({\mathcal H}(X)\). From this the author draws the interesting corollary that \({\mathcal K}(X,M)\) is an ANR.
scientific article; zbMATH DE number 1787527