We first become acquainted with the notion of isotopy of homeomorphisms and with the mapping class group. We continue with a discussion of orientable surfaces and present the classification theorem for compact orientable surfaces. We then introduce cut systems. A cut system is a set of circles embedded in the interior of a surface, along which the surface is cut into smaller pieces. A case of particular interest to us is a pair of pants: any surface homeomorphic to a sphere with three holes. We introduce pants decompositions, which are ways of cutting a surface into pairs of pants. We show how a pants decomposition is encoded by a multigraph and use this to determine which orientable surfaces admit pants decompositions. We then say a few words about moves — ways of passing between different pants decompositions of a given surface. We define the pants graph and describe the action of the mapping class group on its vertices. Finally, we prove a surprising theorem establishing a connection between the orbits of this action and the multigraphs associated to the pants decompositions of the surface.
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