Classificació diferenciable de les superfícies compactes

Abstract

[en] The classification of compact surfaces was already adressed in the 19th century. Both A. F. Möbius and C. Jordan studied this question by considering two surfaces equivalent if they could be decomposed into infinite small pieces such that contiguous pieces of one surface corresponded to contiguous pieces of the other one. In addition, B. Riemann introduced the idea of classifying surfaces according to connectivity. Given a surface, he defined connectivity in base of the maximum number of cuts along closed curves or along arcs joining points on the edge that can be made without making the surface disconnected. The result that determines the classification of compact surfaces is as follows: Every compact, connected surface is diffeomorphic to a unique type of model surface. In this paper we will study the concepts and the theory behind the classification of compact surfaces which are needed to be able to give a proof of the result previously announced. We will go over: Morse functions and Morse’s lemma, the regular interval theorem, isotopy extensions, isotopies of disks and the construction of model surfaces.