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).