On multiplicity of mappings between surfaces

TL;DR

This paper calculates the minimal multiplicity of continuous maps between closed surfaces, relating it to algebraic and topological invariants, and establishes bounds for maps with zero absolute degree.

Contribution

It provides explicit formulas for the minimal multiplicity of maps between surfaces based on their algebraic and topological properties, including cases with zero absolute degree.

Findings

Calculated MMR[f] for maps with positive absolute degree.

Established bounds for MMR[f] when absolute degree is zero.

Expressed MMR[f] in terms of Euler characteristics and fundamental group indices.

Abstract

Let M and N be two closed (not necessarily orientable) surfaces, and f a continuous map from M to N. By definition, the minimal multiplicity MMR[f] of the map f denotes the minimal integer k having the following property: f can be deformed into a map g such that the number |g^{-1}(c)| of preimages of any point c in N under g is at most k. We calculate MMR[f] for any map $f$ of positive absolute degree A(f). The answer is formulated in terms of A(f), [pi_1(N):f_#(pi_1(M))], and the Euler characteristics of M and N. For a map f with A(f)=0, we prove the inequalities 2 <= MMR[f] <= 4.