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.

As said in the lead a pair of pants is any surface which is homeomorphic to a sphere with three holes, which formally are three disjoint open disks removed from the sphere. Thus a pair of pants is a compact surface of genus zero with boundary, the latter being three circles.

The Euler characteristic of a pair of pants is equal to -1. Among all surfaces of negative Euler characteristic it has the maximal one; 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 ${\displaystyle S}$ of genus ${\displaystyle g}$ with ${\displaystyle k}$ boundary components to be ${\displaystyle \xi (S)=3g-3+k}$, 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 ${\displaystyle S}$ and any simple closed curve ${\displaystyle c}$ on ${\displaystyle S}$ which is not homotopic to a boundary component, the compact surface obtain by cutting ${\displaystyle S}$ along ${\displaystyle c}$ has a complexity that is strictly less than ${\displaystyle S}$. In this sense pairs of pants are the only "irreducible" surfaces among all surfaces of negative Euler characteristic.

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