$\pi_1$-injective proper maps between non-compact surfaces

TL;DR

This paper classifies all proper maps between non-compact surfaces that preserve the fundamental group, up to proper homotopy, providing a comprehensive understanding of their structure.

Contribution

It offers a complete classification of $$-injective proper maps between non-compact surfaces, a novel result in geometric topology.

Findings

Complete classification of $$-injective proper maps

Identification of conditions for proper homotopy equivalence

Framework for analyzing maps between non-compact surfaces

Abstract

We classify all $\pi_1$-injective proper maps between non-compact surfaces up to proper homotopy.