Maximal degree of a map of surfaces

TL;DR

This paper establishes a fundamental inequality relating the Euler characteristics of surfaces connected by a map of positive degree, utilizing the Kneser-Edmonds factorization theorem for a natural proof.

Contribution

It proves a new inequality linking surface Euler characteristics via map degree and clarifies the concept of geometric degree for surfaces.

Findings

Proves that hi(M) hi(N)  for maps of degree d>0.

Provides a detailed discussion on the properties of geometric degree.

Includes a natural proof of the Kneser-Edmonds factorization theorem.

Abstract

Given closed possibly nonorientable surfaces $M,N$, we prove that if a map $f:M\to N$ has degree $d>0$, then $\chi(M)\le d\cdot\chi(N)$. We give all necessary comments on the definition and properties of geometric degree, which can be defined for any map. Our proof is based on the Kneser-Edmonds factorization theorem, simple natural proof of which is also presented.