Inmathematics, apair of pants is asurface which ishomeomorphic to the three-holedsphere. The name comes from considering one of the removeddisks as the waist and the two others as the cuffs of apair of pants.

Pairs of pants are used as building blocks forcompact surfaces in various theories. Two important applications are tohyperbolic geometry, where decompositions ofclosed surfaces into pairs of pants are used to construct theFenchel-Nielsen coordinates onTeichmüller space, and intopological quantum field theory where they are the simplest non-trivialcobordisms between 1-dimensionalmanifolds.

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 threeopen disks with pairwise disjoint closures. Thus a pair of pants is a compact surface ofgenus zero with threeboundary components.

TheEuler characteristic of a pair of pants is equal to −1, and the only other surface with this property is the puncturedtorus (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 aconnected compact surface${\displaystyle S}$ ofgenus${\displaystyle g}$ 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 aredisjoint unions of pairs of pants. Furthermore, for any surface${\displaystyle S}$ and anysimple closed curve${\displaystyle c}$ on${\displaystyle S}$ which is nothomotopic to a boundary component, the compact surface obtained 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.

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 apants decomposition for the surface, and the curves are called thecuffs of the decomposition. This decomposition is not unique, but by quantifying the argument one sees that all pants decompositions of a given surface have the same number of curves, which is exactly the complexity.^[1] For connected surfaces a pants decomposition has exactly${\displaystyle 2g-2+k}$ pants.

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.

A given surface has infinitely many distinct pants decompositions (we understand two decompositions to be distinct when they are not homotopic). One way to try to understand the relations between all these decompositions is thepants complex associated to the surface. This is agraph with vertex set the pants decompositions of${\displaystyle S}$, and two vertices are joined if they are related by an elementary move, which is one of the two following operations:

• take a curve${\displaystyle \alpha }$ in the decomposition in a one-holed torus and replace it by a curve in the torus intersecting it only once,
• take a curve${\displaystyle \alpha }$ in the decomposition in a four-holed sphere and replace it by a curve in the sphere intersecting it only twice.

The pants complex isconnected^[2] (meaning any two pants decompositions are related by a sequence of elementary moves) and has infinitediameter (meaning that there is no upper bound on the number of moves needed to get from one decomposition to the other). In the particular case when the surface has complexity 1, the pants complex isisomorphic to theFarey graph.

Theaction of themapping class group on the pants complex is of interest for studying this group. For example, Allen Hatcher and William Thurston have used it to give a proof of the fact that it isfinitely presented.

The interesting hyperbolic structures on a pair of pants are easily classified.^[3]

For all${\displaystyle {\ell }_1,{\ell }_2,{\ell }_3\in (0,\mathrm{\infty })}$ there is a hyperbolic surface${\displaystyle M}$ which is homeomorphic to a pair of pants and whose boundary components are simple closed geodesics of lengths equal to${\displaystyle {\ell }_1,{\ell }_2,{\ell }_3}$. Such a surface is uniquely determined by the${\displaystyle {\ell }_i}$ up toisometry.

By taking the length of a cuff to be equal to zero, one obtains acompletemetric on the pair of pants minus the cuff, which is replaced by acusp. This structure is of finite volume.

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. These arcs are called theseams of the pants.

Cutting the pants along the seams, one gets two right-angled hyperbolic hexagons which have three alternate sides of matching lengths. The following lemma can be proven with elementary hyperbolic geometry.^[4]

If two right-angled hyperbolic hexagons each have three alternate sides of matching lengths, then they are isometric to each other.

So we see that the pair of pants is thedouble of a right-angled hexagon along alternate sides. Since the isometry class of the hexagon is also uniquely determined by the lengths of the remaining three alternate sides, the classification of pants follows from that of hexagons.

When a length of one cuff is zero one replaces the corresponding side in the right-angled hexagon by an ideal vertex.

A point in the Teichmüller space of a surface${\displaystyle S}$ is represented by a pair${\displaystyle (M,f)}$ where${\displaystyle M}$ is a complete hyperbolic surface and${\displaystyle f:S\to M}$ a diffeomorphism.

If${\displaystyle S}$ has a pants decomposition by curves${\displaystyle {\gamma }_i}$ then one can parametrise Teichmüller pairs by the Fenchel-Nielsen coordinates which are defined as follows. Thecuff lengths${\displaystyle {\ell }_i}$ are simply the lengths of the closed geodesics homotopic to the${\displaystyle f({\gamma }_i)}$.

Thetwist parameters${\displaystyle {\tau }_i}$ are harder to define. They correspond to how much one turns when gluing two pairs of pants along${\displaystyle {\gamma }_i}$: this defines them modulo${\displaystyle {\ell }_i\mathbb{Z}}$. One can refine the definition (using either analytic continuation^[5] or geometric techniques) to obtain twist parameters valued in${\displaystyle \mathbb{R}}$ (roughly, the point is that when one makes a full turn one changes the point in Teichmüller space by precomposing${\displaystyle f}$ with aDehn twist around${\displaystyle {\gamma }_i}$).

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. It is aquasi-isometry when Teichmüller space is endowed with theWeil-Petersson metric, which has proven useful in the study of this metric.^[6]

These structures correspond toSchottky groups on two generators (more precisely, if the quotient of thehyperbolic 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 cobordism between twon-dimensionalclosed manifolds is a compact (n+1)-dimensional manifold whose boundary is the disjoint union of the two manifolds. Thecategory of cobordisms of dimensionn+1 is the category with objects the closed manifolds of dimensionn, andmorphisms the cobordisms between them (note that the definition of a cobordism includes the identification of the boundary to the manifolds). Note that one of the manifolds can be empty; in particular a closed manifold of dimensionn+1 is viewed as anendomorphism of theempty set. One can also compose two cobordisms when the end of the first is equal to the start of the second. A n-dimensional topological quantum field theory (TQFT) is a monoidal functor from the category ofn-cobordisms to the category of complex vector space (where multiplication is given by the tensor product).

In particular, cobordisms between 1-dimensional manifolds (which are unions of circles) are compact surfaces whose boundary has been separated into two disjoint unions of circles. Two-dimensional TQFTs correspond toFrobenius 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.

• Hatcher, Allen; Thurston, William (1980). "A presentation for the mapping class group of a closed orientable surface".Topology.19 (3):221–237.doi:10.1016/0040-9383(80)90009-9.
• Imayoshi, Yôichi; Taniguchi, Masahiko (1992).An introduction to Teichmüller spaces. Springer. pp. xiv+279.ISBN 4-431-70088-9.
• Ratcliffe, John (2006).Foundations of hyperbolic manifolds, Second edition. Springer. pp. xii+779.ISBN 978-0387-33197-3.

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