Minimal Simplicial Degree $d$ Maps from Genus $g$ Surfaces to the Torus

Abstract

The degree of a map between orientable manifolds is a fundamental concept in topology, offering deep insights into the structure of the manifolds and the nature of the corresponding maps. This concept has been extensively studied, particularly in the context of simplicial maps between orientable triangulable spaces. In 1982, Gromov proved that if degree $d$ maps exist from a genus $g$ orientable surface to a genus $h$ orientable surface for every $d \in \mathbb{Z}$, then $h$ must be 0 or 1. Recently, degree $d$ self-maps on spheres, particularly on genus 0 surfaces, have been investigated. In this paper, we focus on the unique minimal 7-vertex triangulation of the torus. We construct simplicial degree $d$ maps from a triangulation of a genus $g$ surface to the 7-vertex triangulation of the torus for $g \geq 1$. Our construction of degree $d$ maps is minimal for every $d$ when $g = 1,2$. If $g \geq 3$, then our construction remains minimal for $|d| \geq 2g - 1$. We believe that this concept will be highly useful in combinatorial topology, as it leads to several intriguing open research problems. In the final section, we propose some of these open research problems.