By S. T Hu

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6 Arcs in a Polygon Let P, Q, R, S be points in cyclic order on the boundary of a polygon 9 and let al, a2 be disjoint simple arcs which lie in int(P) except that a1 begins at P and a2 ends at R. Then Q and S are not separated by at u a2 in Y. 2(2). We now pave R2 with rectangular "bricks" of diameter < 6/2 in the pattern shown in Figure 39. 3 The Jordan Curve Theorem 33 w I I I Figure 39 P Figure 40 through a corner or touch (as distinct from cross) an edge of a brick, the same is true for d n 9.

Since all the cell decompositions we use can be viewed in this way, it will not be necessary to make our definitions of cell - complex and elementary subdivision any more formal, since in the last resort one can always view cells and the dividing cells inside them as unions of simplexes in a simplicial decomposition. The point of considering cell complexes at all is to minimize the number of cells, which usually helps to shorten computations. Obtain the two decompositions of the torus in Figure 29 by elementary subdivision of the square cell structure.

An example of what a coherent orientation for a 2-manifold looks like is given in Figure 24. Intuitively, one can slide a circular arrow all over the surface and match it (Po, P1, P2) = (P1, P2, Po) = (P2, Po, Pt) 2 Pn (Po, P2, Pi) = (P2, P11 Po) = (PI, Po, P2) Figure 23 Figure 24 0 Introduction and Foundations 22 Figure 25 with the circular arrow drawn in each triangle. A complex is called orientable if it has a coherent orientation. The classic nonorientable figure is the Mobius band (Figure 25).

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