A caracter\'istica de Euler-Poincar\'e

TL;DR

This paper introduces the Euler-Poincaré characteristic in an elementary and historical manner, discussing its applications to various types of surfaces and potential extensions to higher dimensions.

Contribution

It offers a simplified, historical explanation of the Euler-Poincaré characteristic and discusses its applicability to different surface types, including singular and non-oriented surfaces.

Findings

Introduces the Euler-Poincaré characteristic in an elementary way

Discusses its application to smooth, singular, and non-oriented surfaces

Explores potential extensions to higher-dimensional spaces

Abstract

We introduce the Euler-Poincar\'e's characteristic with an elementary way and historically. We explain also why one should call Descartes-Poincar\'e characteristic instead of the Euler-Poincar\'e's characteristic. All the considered spaces are compact and without boundary. We work essentially on smooth and oriented surfaces . However, we work also on singular and non-oriented surfaces. We provide also the results which may be extended for the case of superior dimension.