In this talk, we discuss proper maps between two non-compact surfaces, with a particular emphasis on facts stemming from two fundamental questions in topology: whether every homotopy equivalence between two $n$-manifolds is homotopic to a homeomorphism, and whether every degree-one self-map of an oriented manifold is a homotopy equivalence.

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, orientable surfaces are properly rigid. In fact, we prove a stronger result: if a homotopy equivalence between any two noncompact, orientable 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 $\pi_1$-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 $f\colon M\to M$ of degree one is a homotopy equivalence. This is the natural topological analog of Hopfian groups. H. Hopf posed the question of 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.