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).
Then the only surfaces with negative Euler characteristic and complexity zero are disjoint unions of pairs of pants.
which is not homotopic to a boundary component, the compact surface obtained by cutting
A collection of simple closed curves on a surface is a pants decomposition if and only if they are disjoint, no two of them are homotopic and none is homotopic to a boundary component, and the collection is maximal for these properties.
One way to try to understand the relations between all these decompositions is the pants complex associated to the surface.
, and two vertices are joined if they are related by an elementary move, which is one of the two following operations: The pants complex is connected[2] (meaning any two pants decompositions are related by a sequence of elementary moves) and has infinite diameter (meaning that there is no upper bound on the number of moves needed to get from one decomposition to the other).
For example, Allen Hatcher and William Thurston have used it to give a proof of the fact that it is finitely presented.
The interesting hyperbolic structures on a pair of pants are easily classified.
The geometric proof of the classification in the previous paragraph is important for understanding the structure of hyperbolic pants.
It proceeds as follows: Given a hyperbolic pair of pants with totally geodesic boundary, there exist three unique geodesic arcs that join the cuffs pairwise and that are perpendicular to them at their endpoints.
Cutting the pants along the seams, one gets two right-angled hyperbolic hexagons which have three alternate sides of matching lengths.
[4] So we see that the pair of pants is the double of a right-angled hexagon along alternate sides.
When a length of one cuff is zero one replaces the corresponding side in the right-angled hexagon by an ideal vertex.
then one can parametrise Teichmüller pairs by the Fenchel-Nielsen coordinates which are defined as follows.
One can refine the definition (using either analytic continuation[5] or geometric techniques) to obtain twist parameters valued in
One can define a map from the pants complex to Teichmüller space, which takes a pants decomposition to an arbitrarily chosen point in the region where the cuff part of the Fenchel-Nielsen coordinates are bounded by a large enough constant.
[6] These structures correspond to Schottky groups on two generators (more precisely, if the quotient of the hyperbolic plane by a Schottky group on two generators is homeomorphic to the interior of a pair of pants then its convex core is an hyperbolic pair of pants as described above, and all are obtained as such).
A n-dimensional topological quantum field theory (TQFT) is a monoidal functor from the category of n-cobordisms to the category of complex vector space (where multiplication is given by the tensor product).
Two-dimensional TQFTs correspond to Frobenius algebras, where the circle (the only connected closed 1-manifold) maps to the underlying vector space of the algebra, while the pair of pants gives a product or coproduct, depending on how the boundary components are grouped – which is commutative or cocommutative.
Further, the map associated with a disk gives a counit (trace) or unit (scalars), depending on grouping of boundary, which completes the correspondence.
