By Shiing-Shen Chern
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Extra resources for Complex Manifolds
1 when demonstrating the no/j-orientability of Κ may be described on the plane model of Κ as illustrated in Fig. 1. This demonstration shows that Κ is /jon-orientable essentially because its surface contains a Mobius band along which the path is taken: see Fig. 2. In the light of this insight, let us consider the orientability of our fourth basic surface, the projective plane. As shown in Fig. 3, it is simple to trace out a Mobius band on a plane model for P. It follows that the projective plane is also a non-orientable compact surface.
2 illustrates the process of obtaining a space model of 2Γ from this plane model. It is a simple matter to extend the above method to the construction of a plane model of the connected sum of any number of basic surfaces. In Fig. 3 we apply the procedure to the surface T#K#P#T. ) We take the component plane models, line them up, join them up, and open out the resulting diagram. Surfaces as plane diagrams 50 Vertices on plane nnodel (a) (b) (d) Fig. 1. 3 Sec. 21 Algebraic description of surfaces 51 Fig.
It will prove to be convenient to add a further refinement to our models by making the labels attached to the edges carry the information given by the arrows. Sec. 1] The basic plane models Torus with two loop cuts/i and Fig. 2 . 1 . 1 . Fig. 2. 2 We do this by assigning an underlying direction to the boundary of the rectangle; say clockwise. Then we attach an index - 1 to the label of any edge whose arrow points in the opposite direction. Applying this procedure to the rectangle of Fig. 2. we obtain the diagram shown in Fig.