There are two important groups associated with a closed surface: the fundamental group and the mapping class group. The former, also known as the first homotopy group of the surface (considered as a topological space), is probably the more familiar of the two. It was defined and explored by Henri Poincaré . In this chapter I would like to revive a few of the traditional images that go with the second, more picturesque group. Customary definitions of these two groups go something like this.
KeywordsFundamental Group Braid Group Height Function Mapping Class Group Invariant Circle
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