Pair of pants (mathematics)

In mathematics, a pair of pants is a surface which is homeomorphic to the three-holed sphere. The name comes from considering one of the removed disks as the waist and the two others as the cuffs of a pair of pants. Pairs of pants are used as building blocks for compact surfaces in various theories. Two important applications are to hyperbolic geometry, where decompositions of closed surfaces into pairs of pants are used to construct the Fenchel-Nielsen coordinates on Teichmüller space, and in topological quantum field theory where they are the simplest non-trivial cobordisms between 1-dimensional manifolds. A pair of pants is any surface that is homeomorphic to a sphere with three holes, which formally is the result of removing from the sphere three open disks with pairwise disjoint closures. Thus a pair of pants is a compact surface of genus zero with three boundary components. The Euler characteristic of a pair of pants is equal to −1, and the only other surface with this property is the punctured torus (a torus minus an open disk). The importance of the pairs of pants in the study of surfaces stems from the following property: define the complexity of a connected compact surface of genus with boundary components to be , and for a non-connected surface take the sum over all components. Then the only surfaces with negative Euler characteristic and complexity zero are disjoint unions of pairs of pants. Furthermore, for any surface and any simple closed curve on which is not homotopic to a boundary component, the compact surface obtained by cutting along has a complexity that is strictly less than . In this sense, pairs of pants are the only "irreducible" surfaces among all surfaces of negative Euler characteristic. By a recursion argument, this implies that for any surface there is a system of simple closed curves which cut the surface into pairs of pants. This is called a pants decomposition for the surface, and the curves are called the cuffs of the decomposition.