On the concept of genus in topology and complex analysis
Friedrich E. P. Hirzebruch, Matthias Kreck · MPG.PuRe (Max Planck Society) · 2009
biology, where it is used to connote a grouping of organisms having common characteristics. In mathematics the word is also used to group objects with common characteristics. The concept of genus arises in various mathematical contexts, such as number theory, as well as in the areas we consider in this article, topology and complex analysis. Even within the latter two areas there are various notions of genus that historically originated with the genus of an oriented surface. We begin with these origins and afterward treat generalizations and modifications. We provide no detailed definitions and proofs; rather, our goal is to give the reader an intuitive feeling for the concept of genera. The Genus of a Surface In his paper “Theorie der Abel’schen Functionen” [20] Riemann studied the topology of surfaces. He classified a surface by looking for simple closed curves along which to cut in order to obtain a simple presentation of the surface. He called the minimal number of such curves 2p and showed that this invariant determines the surface. A few years later, when Clebsch studied surfaces from a more algebraic geometric viewpoint, he calledp “das Geschlecht (genus) ” of the surface. In more modern terms one can formulate Riemann’s insight as follows: every connected, closed Friedrich E. P. Hirzebruch is professor emeritus at the