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 Nielsenâ€™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.

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Last updated: 15 Jul 2024