The Effect of Singularization on the Euler Characteristic

TL;DR

This paper investigates how certain operations on smooth surfaces create singular surfaces and derives formulas relating their Euler characteristic and genus, extending classical concepts to singular cases.

Contribution

It introduces a formula for the Euler characteristic of singularized surfaces and a new definition of genus that generalizes the classical one.

Findings

Derived a formula for Euler characteristic of singularized surfaces

Introduced a generalized genus concept for singular surfaces

Proved a relation between Euler characteristic and genus in singular cases

Abstract

In this work, singular surfaces are obtained from smooth orientable closed surfaces by applying three basic simple loop operations, collapsing operation, zipping operation and double loop identification, each of which produces different singular surfaces. A formula that provides the Euler characteristic of the singularized surface is proved. Also, we introduce a new definition of genus for singularized surfaces which generalizes the classical definition of genus in the smooth case. A theorem relating the Euler characteristic to the genus of the singularized surface is proved.