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Explorations in Topology by David Gay

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6

Doughnuts

This chapter introduces the torus, the surface of a doughnut. Maps and networks are considered on this surface. The analog to Euler’s formula for maps on a sphere is conjectured and proved for the torus. The five-terminal problem (Can the complete graph on five points K5 be drawn?), previously investigated for the sphere and shown to be impossible there, is shown to be possible on the torus. A consequence of this is the presentation of a map on the torus requiring five colors. The theme of comparing torus with sphere is pursued in the end-of-chapter investigations where additional problems with maps and networks once investigated for the sphere are considered on the torus. In the notes, the Jordan curve theorem for the sphere is discussed ...

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