Classification of compact real surfaces: a quick proof

TL;DR

This paper presents a concise algebraic proof of the classification theorem for compact real surfaces, simplifying the traditional complex proofs based on surgery and visual methods.

Contribution

It introduces a quick algebraic approach to classify compact real surfaces, offering a more straightforward alternative to classical proofs.

Findings

Provides a simplified algebraic proof of the classification theorem.

Reduces reliance on surgery and pictorial methods.

Enhances understanding of surface classification through algebraic techniques.

Abstract

The aim of this note is to give a quick algebraic proof of (the combinatorial part of) the classification theorem for compact real surfaces, whose classical proofs (as in the Massey book and in the Conway ZIP proof) are based on surgery (and pictures) and look more intricate.