Riemann Surfaces

Abstract

Turning aside from the rather general topics treated in the preceding part, consider now the special case of a compact connected Riemann surface M. Some familiarity with the topology of surfaces will be presupposed; so it can be taken as known that topologically M is a sphere with g handles, where the integer g is called the genus of the surface. The surface M can then be dissected into a contractible set by cutting along 2g paths which issue from a common base point p₀∊M and are otherwise disjoint, one pair of paths for each handle, as illustrated in Fig. 1 ; note that these paths are viewed as oriented paths, with the orientations as indicated by the directions of the arrows. The complement of this set of paths is the interior of an oriented polygonal region Δ having 4g boundary arcs, one pair of boundary arcs corresponding to the two sides of each path of the dissection of M, as illustrated in Fig. 2; these boundary arcs inherit orientations from the orientations of the paths of the dissection, and the oriented boundary of Δ is then the chain

$$\partial \Delta = \sum\nolimits_{{i = 1}}^{g} {(\alpha _{i}^{'} + \beta _{i}^{'} - \alpha _{i}^{'} - \beta _{i}^{})}$$

. The choice of a base point p_o∊M and a canonical dissection by such oriented paths α₁,...,α _g , β₁,..., β _g is called a marking of the surface M, and a surface with a particular marking is called a marked surface. A Riemann surface has a canonical orientation, and markings of a Riemann surface will be restricted to those for which the orientation of Δ is the canonical orientation. There are of course a great many possible markings for any Riemann surface M.