Minimal Simplicial Degree $d$ Maps from Genus $g$ Surfaces to the Torus
Biplab Basak, Ayushi Trivedi
TL;DR
The paper addresses the problem of realizing minimal degree $d$ simplicial maps from triangulated genus $g$ surfaces to the unique $7$-vertex triangulation of the torus. It constructs degree $d$ maps by triangulating $\Sigma_g$ with $n=7d+2-2g$ vertices and mapping $u_{(i,j)}$ to $v_i$, ensuring a consistent degree via algebraic counting of preimages. It proves optimality results: the construction is minimal for $g=1,2$ for all $d$, and minimal for $|d|\ge 2g-1$ when $g\ge 3$; it also provides a general connected-sum framework to obtain degree $d$ maps $\Sigma_{g_1}\to \Sigma_{g_2}$ with $|d|\le \left\lfloor\frac{g_1-1}{g_2-1}\right\rfloor$. The work advances combinatorial topology by linking explicit triangulations, degree theory, and simplicial volume considerations, and it opens multiple open problems and directions for higher-dimensional analogues and alternative target triangulations.
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.