Comptes Rendus. Mathématique (Oct 2023)
Surfaces of infinite-type are non-Hopfian
Abstract
We show that finite-type surfaces are characterized by a topological analogue of the Hopf property. Namely, an oriented surface $\Sigma $ is of finite-type if and only if every proper map $f\colon \,\Sigma \rightarrow \Sigma $ of degree one is homotopic to a homeomorphism.