The classification of surfaces via normal curves

TL;DR

This paper offers a straightforward proof of the surface classification theorem utilizing normal curves, avoiding algebraic topology invariants, and drawing parallels to prime decomposition proofs in 3-manifolds.

Contribution

It introduces a simple, invariant-free proof of the surface classification theorem based on normal curves, inspired by 3-manifold prime decomposition methods.

Findings

Proof simplifies surface classification understanding

Avoids use of algebraic topology invariants

Establishes analogy with 3-manifold prime decomposition

Abstract

We present a simple proof of the surface classification theorem using normal curves. This proof is analogous to Kneser's and Milnor's proof of the existence and uniqueness of the prime decomposition of 3-manifolds. In particular, we do not need any invariants from algebraic topology to distinguish surfaces.