Maps Between Non-compact Surfaces

Abstract

This thesis focuses on studying proper maps between two non-compact surfaces with a particular emphasis on the topological rigidity and the Hopfian property. Topological rigidity is the property that every homotopy equivalence between two closed n-manifolds is homotopic to a homeomorphism. This property refines the notion of homotopy equivalence, implying homeomorphism for a particular class of spaces. According to Nielsen’s results from the 1920s, compact surfaces exhibit topological rigidity. However, topological rigidity fails in dimensions three and above, as well as for compact bordered surfaces. We prove that all non-compact surfaces are properly rigid. In fact, we prove a stronger result: if a homotopy equivalence between any two non-compact surfaces is a proper map, then it is properly homotopic to a homeomorphism, provided that the surfaces are neither the plane nor the punctured plane. As an application, we also prove that any π₁-injective proper map between two non-compact surfaces is properly homotopic to a finite-sheeted covering map, given that the surfaces are neither the plane nor the punctured plane. An oriented manifold M is said to be Hopfian if every self-map f: M → M of degree one is a homotopy equivalence. This is the natural topological analogue of Hopfian groups. H. Hopf questioned whether every closed, oriented manifold is Hopfian. We prove that every oriented infinite-type surface is non-Hopfian. Consequently, an oriented surface S is of finite type if and only if every proper self-map of S of degree one is homotopic to a homeomorphism.