Surfaces of infinite-type are non-Hopfian

TL;DR

This paper characterizes finite-type surfaces through a topological property analogous to Hopf's, showing that such surfaces are exactly those where degree-one proper maps are homotopic to homeomorphisms.

Contribution

It establishes a new topological characterization of finite-type surfaces using a Hopf-like property, linking proper maps and homotopy to homeomorphisms.

Findings

Finite-type surfaces are characterized by a Hopf-like property.

Proper degree-one maps on finite-type surfaces are homotopic to homeomorphisms.

Surfaces of infinite type do not satisfy this property.

Abstract

We show that finite-type surfaces are characterized by a topological analog of the Hopf property. Namely, an oriented surface $\Sigma$ is of finite-type if and only if every proper map $f\colon\Sigma\to \Sigma$ of degree one is homotopic to a homeomorphism.