Curves on Surfaces
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The fundamental problem in the topological classification of curves on surfaces is to decide whether a given closed curve contracts to a point. We shall call this the contractibility problem. Jordan 1866b recognized that the problem could be expressed in algebraic terms, but his work contained errors. He showed that each curve could be deformed into a product of certain canonical curves—essentially the generators of the fundamental group—and realized that the canonical curves satisfied certain relations. However, he seemed not to notice that he actually had a group (surprisingly, in view of the subsequent appearance of his pioneering treatise on group theory, Jordan 1870), and failed to get the right relations.
KeywordsWord Problem Homology Class Mapping Class Group Klein Bottle Simple Closed Curve
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