Homology of polyhedra and quadrangulations of surfaces

TL;DR

This paper introduces a new algebraic topology formula involving Betti numbers and develops the spinal method for generating quadrangulations of closed orientable surfaces derived from polyhedral spines in Euclidean space.

Contribution

It presents a novel formula in algebraic topology and a new spinal method for constructing quadrangulations of surfaces from polyhedral spines.

Findings

Derived a new Betti number formula in algebraic topology.

Developed the spinal method for surface quadrangulations.

Constructed quadrangulations with specific properties using spinal manipulation.

Abstract

A new formula is obtained in algebraic topology, in terms of Betti numbers, and a new method, called the spinal method, is suggested and developed for generating quadrangulations of closed orientable surfaces. Those surfaces arise as the thickenings of 1- and 2-dimensional curvilinear polyhedra, called spines, in Euclidean 3-space. By way of spinal manipulation, quadrangulations with given properties are constructed.