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Which is the word representation of an m-holed torus. If m = 0, then Mo = w W - I , which is the word representation of a sphere. We have thus converted Mo into the word representation of either the surface m T or the surface S by the use of Operations 1,2, and 3, each of which is reversible. By carrying out the sequence of operations in reverse order, we can convert our final word representation of a surface into the original word representation Mo of an orientable 2n-gon. Since the operations do not alter, up to homeomorphism, the surface represented, it follows that Mo is the word representation of an orientable compact surface.

We choose the discs so that C I and C2 pass through points of the surfaces Twhich are represented by the vertices PI and P2 of the plane models. This is shown in Fig. 1(a). After we have have cut out the interiors of the discs, the resulting sets are represented in the plane by the pentagons of Fig. 1 (b). Finally, we identify the sides C I and C 2 of the pentagons, as shown in Fig. 1(c). This process corresponds to the identification of the perimeters of C I and C2 on the space models. We end up with the oc'agonal plane model of 2Tshown in Fig.

These facts, together with the commutativity and associativity of the connected sum operation, are useful in converting an awkward expression for a compact surface into the normal form. 1 Express TPKP(2K)T in normal form. Solution TPKP(2K)T== T(PK)(PK)KT == T(TP)(TP)KT == (4np(PK) ==(4np(TP) == (Sn(2P) ==(SnK. 7 SUMMARY By interpreting the connected sum construction in terms of plane models, we were able to show how to form a plane model of any connected sum of basic surfaces. Assuming the classification theorem, this enabled us to form a plane model of any compact surface.

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